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10.4: Triangles

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    129642
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    A view of an arched ceiling in an architectural building.
    Figure 10.53 The appearance of triangles in buildings is part of modern-day architectural design. (credit: "Inside Hallgrímskirkja church, Reykjavik, Iceland" by O Palsson/Flickr, CC BY 2.0)

    Learning Objectives

    After completing this section, you should be able to:

    1. Identify triangles by their sides.
    2. Identify triangles by their angles.
    3. Determine if triangles are congruent.
    4. Determine if triangles are similar.
    5. Find the missing side of similar triangles.

    How were the ancient Greeks able to calculate the radius of Earth? How did soldiers gauge their target? How was it possible centuries ago to estimate the height of a sail at sea? Triangles have always played a significant role in how we find heights of objects too high to measure or distances between objects too far away to calculate. In particular, the concept of similar triangles has countless applications in the real world, and we shall explore some of those applications in this section.

    Technology has given us instruments that allow us to find measurements of distant objects with little effort. However, it is all based on the properties of triangles discovered centuries ago. In this section, we will explore the various types of triangles and their special properties, as well as how to measure interior and exterior angles. We will also explore congruence theorems and similarity.

    Identifying Triangles

    Joining any three noncollinear points with line segments produces a triangle. For example, given points AFigure 10.54.

    A triangle with points A, B, and C, and sides a, b, and c.
    Figure 10.54 Triangle

    Triangles are classified by their angles and their sides. All angles in an acute triangle measure <90.Figure 10.55 illustrates the shapes of the basic triangles, their names, and their properties.

    A few other facts to remember as we move forward:

    • The points where the line segments meet are called the vertices (plural for vertex).
    • We often refer to sides of a triangle by the angle they are opposite. In other words, side aa is opposite angle AA, side bb is opposite angle BB, and side cc is opposite angle CC.
    Six triangles. An acute triangle. A right triangle. An obtuse triangle. An isosceles triangle has two equal sides and two equal angles. An equilateral triangle has all three sides equal and all three angles are equal. A scalene triangle. All three sides are unequal.
    Figure 10.55 Types of Triangles

    We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side opposite the right angle is its longest side and is called the hypotenuse, and the sides adjacent to the right angle are called the legs.

    One of the most important properties of triangles is that the sum of the interior angles equals 180.Figure 10.56.

    Two horizontal lines intersected by two transversals. The first transversal makes four angles with the bottom line. Two angles are unknown. One of the interior angles is marked 1 and one of the exterior angles is marked A. The second transversal makes four angles with the bottom line. Two angles are unknown. One of the interior angles is marked 3 and one of the exterior angles is marked B. The two transversals meet at a point on the line at the top. Six angles are formed around this intersection point. The interior angles are labeled 2, 5, and 4. Two exterior angles are unknown and the third angle is marked C.
    Figure 10.56 Sum of Interior Angles

    This is how the proof goes:

    Step 1: Start with a straight line ABAB and a point CC not on the line.

    Step 2: Draw a line through point CC parallel to the line ABAB.

    Step 3: Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, to intersect the parallel lines.

    Step 4: Because of the property that alternate interior angles inside parallel lines are equal, we have that

    m2=m1andm3=m4.m2=m1andm3=m4.

    Step 5: Notice that m2+m5+m4=180m2+m5+m4=180 by the straight angle property.

    Step 6: Therefore, by substitution, m1=m2,m1=m2, and m3=m4,m3=m4, we have that

    m1+m3+m5=180.m1+m3+m5=180.

    Therefore, the sum of the interior angles of a triangle=180.triangle=180.

    Example 10.15

    Finding Measures of Angles Inside a Triangle

    Find the measure of each angle in the triangle shown (Figure 10.57). We know that the sum of the angles must equal 180.180.

    A triangle with its angles marked x degrees, (x plus 17) degrees, and (3 x minus 62)
    Figure 10.57
    Answer

    Step 1: As the sum of the interior angles equals 180,180, we can use algebra to find the measures:

    x + ( x + 17 ) + ( 3 x 62 ) = 180 5 x 45 = 180 5 x = 225 x = 225 5 = 45 x + ( x + 17 ) + ( 3 x 62 ) = 180 5 x 45 = 180 5 x = 225 x = 225 5 = 45

    Step 2: Now that we have the value of xx, we can substitute 45 into the other two expressions to find the measure of those angles:

    (x+17)=45+17=62(3x62)=3(45)62=13562=73 (x+17)=45+17=62(3x62)=3(45)62=13562=73

    Step 3: Then, mx=45mx=45, m(x+17)=62m(x+17)=62, and m(3x62)=73.m(3x62)=73.

    Your Turn 10.15

    1.
    Find the measures of each angle in the triangle shown.
    A triangle with its angles marked x degrees, (x plus 1) degrees, and (x minus 19) degrees.
    Figure 10.58

    Example 10.16

    Finding Angle Measures

    Find the measure of angles numbered 1–5 in Figure 10.59.

    Three triangles lie on a horizontal line. The first and third triangles lie above the line and the second triangle lies below the line. The interior angles of the first triangle measure 30 degrees, 89 degrees, and unknown. The exterior angle made by this triangle with the line is labeled 1. The interior angles of the second triangle measure 2, 3, and 4. The interior angles of the third triangle are labeled 5, 75 degrees, and unknown. The exterior angle made by this triangle with the line is labeled 134 degrees.
    Figure 10.59
    Answer

    The m1=119m1=119 because it is supplementary with the unknown angle of the adjacent triangle. The unknown angle measures 61.61. The m2=61m2=61 because of vertical angles. The m5=59m5=59 because the angle that is supplementary to the 134134 measures 4646, and angle 5 is the unknown angle in that triangle. The m4=59m4=59 by vertical angles. Finally, m3=60,m3=60, as it is the third angle in the triangle with angles measuring 5959 and 61.61.

    Your Turn 10.16

    1.
    Find the measure of angles 1, 2, and 3 in the figure shown.
    Two triangles and a line lie on a horizontal line. The first triangle lies above the line and the second triangle lies below the line. The interior angles of the first triangle are labeled 35 degrees, unknown, and unknown. The exterior angle made by this triangle with the line is labeled 145 degrees. The interior angles of the second triangle measure 1, 2, and 3. An increasing line arises from the endpoint of the second triangle and it makes an angle of 60 degrees with the horizontal line.
    Figure 10.60

    Congruence

    If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent.

    Example 10.17

    Determining If Triangles Are Congruent

    In Figure 10.61, is the triangle ABCABC congruent to triangle DEFDEF?

    Two triangles, A B C and D E F. Angles A and C are equal and congruent to angles D and F. The sides, A B and B C are equal. The side, A B is congruent to the side, D E. The sides, D E and E F are equal. The side, B C is congruent to the side, E F.
    Figure 10.61
    Answer

    Triangle ABCABC is congruent to triangle DEFDEF. Angles AA and CC are congruent to angles DD and FF, which implies that angle BB is congruent to angle EE. Side ABAB is congruent to side DEDE, and side CBCB is congruent to side FEFE, which implies that side ACAC is congruent to side DFDF.

    Your Turn 10.17

    1.
    In the figure shown, is triangle /**/ABC/**/ congruent to triangle /**/DEF/**/?
    Two triangles, A B C and D E F. Angles A is congruent to angle D. Angle C is congruent to angle F. The side, A C is congruent to the side, D F. The side, B C is congruent to the side, E F.
    Figure 10.62

    The Congruence Theorems

    The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol to define congruence. For example, ΔABCΔDEFΔABCΔDEF.

    Side-Side-Side (SSS). If three sides of one triangle are equal to the corresponding sides of the second triangle, then the triangles are congruent. See Figure 10.63.

    Two triangles, D E F and R S T. The side, D F is congruent to the side, R T. The side, E F is congruent to the side, S T. The side, D E is congruent to the side, R S.
    Figure 10.63 Side-Side-Side (SSS)

    We have that DF¯RT¯DF¯RT¯, EF¯ST¯,EF¯ST¯, and DE¯RS¯,DE¯RS¯, then ΔDEFΔRST.ΔDEFΔRST.

    Side-Angle-Side (SAS). If two sides of a triangle and the angle between them are equal to the corresponding two sides and included angle of the second triangle, then the triangles are congruent. See Figure 10.64. We see that AB¯AB¯AB¯AB¯ and BC¯BC¯BC¯BC¯, mB=mBmB=mB, then ΔABCΔABCΔABCΔABC.

    Two triangles, A B C and A prime B prime C prime. The angles, B and B prime are congruent. The side, A B is congruent to the side, A prime B prime. The side, B C is congruent to the side, B prime C prime.
    Figure 10.64 Side-Angle-Side (SAS)

    Angle-Side-Angle (ASA). If two angles and the side between them in one triangle are congruent to the two corresponding angles and the side between them in a second triangle, then the two triangles are congruent. See Figure 10.65. Notice that mAmFmAmF, and mCmDmCmD, AC¯DF¯AC¯DF¯, then ΔABCΔDEF.ΔABCΔDEF.

    Two triangles, A B C and D E F. The sides, C A and D F rest on the same line. The sides, C A and D F are equal. The angles, A and F are congruent. The angles, C and D are congruent.
    Figure 10.65 Angle-Side-Angle (ASA)

    Angle-Angle-Side (AAS). If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded corresponding side of a second triangle, then the triangles are congruent.

    See Figure 10.66. We see that mXmXmXmX, mZmZmZmZ, and XY¯XY¯XY¯XY¯, then ΔXYZΔXYZΔXYZΔXYZ.

    Two triangles, X Y Z and X prime Y prime Z prime. In the triangle X Y Z, the side X Y measures 5 centimeters, and the angles X and Z measure 22 degrees and 118 degrees. In the triangle, X prime Y prime Z prime, the side X prime Y prime measures 5 centimeters, and the angles X prime ad Z prime measure 22 degrees and 118 degrees.
    Figure 10.66 Angle-Angle-Side (AAS)

    Example 10.18

    Identifying Congruence Theorems

    What congruence theorem is illustrated in Figure 10.67?

    A rectangle is formed by joining two right triangles. The hypotenuses of both the triangles share the same side. The top-left and bottom-right angles of the rectangle are equal.
    Figure 10.67
    Answer

    AAS: Two angles and a non-included side in one triangle are congruent to the corresponding angles and side in the second triangle.

    Your Turn 10.18

    1.
    Identify the congruence theorem being illustrated in the figure shown.
    Two triangles. The bottom-left of the first triangle is congruent with the top-right angle of the second triangle. The left side of the first triangle is congruent with the right side of the second triangle. The bottom side of the first triangle is congruent with the top side of the second triangle.
    Figure 10.68

    Example 10.19

    Determining the Congruence Theorem

    What congruence theorem is illustrated in Figure 10.69?

    Two triangles. All three sides of each triangle are of different lengths.
    Figure 10.69
    Answer

    The SSS theorem.

    Your Turn 10.19

    1.
    What congruence theorem is being illustrated in the figure shown?
    Two triangles. The top angle of the first triangle and the bottom-left angle of the second triangle are congruent. The bottom-right angle of the first triangle is congruent to the top angle of the second triangle. The right side of the first triangle is congruent with the left side of the second triangle.
    Figure 10.70

    Similarity

    If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction abab is called a ratio, but two fractions equal to each other is called a proportion, such as ab=cd.ab=cd.

    This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a scaling factor, which is the ratio of corresponding measurements between an object or representation of the object, to an image that produces the second, similar image.

    For example, why are the two images in Figure 10.71 are similar? These two images have the same proportions between elements. Therefore, they are similar.

    Two smiley faces. The first one is bigger and the second one is smaller.
    Figure 10.71 Similarity

    Example 10.20

    Determining If Triangles Are Similar

    Are the two triangles shown in Figure 10.72 similar?

    Two right triangles. In the first triangle, the legs measure 4 and 7. The hypotenuse measures 8.06. The top, bottom-left, and bottom-right angles measure 57 degrees, 90 degrees, and 33 degrees. The triangle is labeled alpha. In the second triangle, the legs measure 3.5 and 2. The hypotenuse measures 4.03. The bottom, top-left, and top-right angles measure 57 degrees, 33 degrees, and 90 degrees. The triangle is labeled beta.
    Figure 10.72
    Answer

    Step 1: We will look at the proportions within each triangle. In triangle αα (alpha), the side opposite the 5757 angle measures 7, and the side opposite the 3333 angle measures 4. Then, the measures of the corresponding sides in triangle ββ (beta) measures 3.5 and 2, respectively. We have

    4 7 = 0.5714 2 3.5 = 0.5714. 4 7 = 0.5714 2 3.5 = 0.5714.

    This is the proportion 47=23.547=23.5. The scaling factor is 0.5714.

    Step 2: Let’s try another correspondence. In triangle αα, the hypotenuse measures 8.06 and the side opposite the 5757 angle measures 7. In triangle ββ, the hypotenuse measure 4.03 and the side opposite the 5757 angle measures 3.5. We have

    7 8.06 = 0.8685 3.5 4.03 = 0.8685. 7 8.06 = 0.8685 3.5 4.03 = 0.8685.

    Step 3: Now, let’s look at the proportions between triangle αα and triangle β.β. The side measuring 2 in triangle ββ corresponds to the side measuring 4 in triangle αα, the side measuring 3.5 in triangle ββ corresponds to the side measuring 7 in triangle α,α, and the hypotenuse in triangle ββ corresponds to the hypotenuse in triangle α.α. We have

    2 4 = 0.5 3.5 7 = 0.5 4.03 8.06 = 0.5 2 4 = 0.5 3.5 7 = 0.5 4.03 8.06 = 0.5

    Thus, the corresponding angles are equal and the proportions between each pair of corresponding sides equals 0.5. In other words, the scaling factor is 0.5. Therefore, the triangles are similar.

    Your Turn 10.20

    1.
    Is triangle /**/EFG/**/ similar to triangle /**/JKL/**/ in the figure shown?
    Two right triangles, E F G and J K L. In the first triangle E F G, the legs G E and E F measure 3.7 and 3.4. The hypotenuse G F measures 5. In the second triangle J K L, the legs L J and J K measure 4.9 and 4.5. The hypotenuse L K measures 6.6. The angles, G and L are congruent.
    Figure 10.73

    Example 10.21

    Proving Similarity

    In Figure 10.74, is triangle δδ (delta) similar to triangle εε (epsilon)? Find the lengths of sides xx and yy as part of your answer.

    Two right triangles are labeled delta and epsilon. In the first triangle, the legs measure x and 2.375. The hypotenuse measures 6. In the second triangle, the legs measure 2.475 and 1.069. The hypotenuse measures y. The angles in both the triangles are congruent.
    Figure 10.74
    Answer

    We can see that all three angles in triangle δδ are equal to the corresponding angles in triangle εε. That is enough to determine similarity. However, we want to find the values of xx and yy to prove similarity.

    Step 1: We have to do is set up the proportions between the corresponding sides. We have the side that measures 2.375 in triangle δδ corresponding to the side measuring 1.069 in triangle εε. We have the hypotenuse/side in triangle δδ measuring 6 corresponding to the hypotenuse/side labeled yy in triangle εε. And, finally, the side labeled xx in triangle δδ corresponds to the side measuring 2.475 in triangle ε.ε.

    Each proportion should be equal. We start with the proportion of the shorter sides. Thus

    1.069 2.375 = 0.45 1.069 2.375 = 0.45

    Step 2: We solve for yy using the first proportion. Set the two ratios equal to each other, cross-multiply, and solve for yy. We have:

    1.069 2.375 = y 6 ( 6 ) ( 1.069 ) = ( 2.375 ) ( y ) 6.414 = 2.375 y 6.414 2.375 = 2.7 = y 1.069 2.375 = y 6 ( 6 ) ( 1.069 ) = ( 2.375 ) ( y ) 6.414 = 2.375 y 6.414 2.375 = 2.7 = y

    So, y=2.7y=2.7.

    Step 3: Checking that length in the proportion factor of 0.45, we have:

    y 6 = 2.7 6 = 0.45 y 6 = 2.7 6 = 0.45

    Step 4: Solving for xx, we will use the same proportion we used to solve for yy. We have:

    1.069 2.375 = 2.475 x 1.069 ( x ) = 2.475 ( 2.375 ) 1.069 ( x ) = 5.878 x = 5.878 1.069 = 5.5 2.475 x = 2.475 5.5 = 0.45 1.069 2.375 = 2.475 x 1.069 ( x ) = 2.475 ( 2.375 ) 1.069 ( x ) = 5.878 x = 5.878 1.069 = 5.5 2.475 x = 2.475 5.5 = 0.45

    Step 5: We test the proportions. We have the following:

    2.7 6 = 2.475 5.5 = 1.069 2.375 = 0.45 2.7 6 = 2.475 5.5 = 1.069 2.375 = 0.45

    The proportions are all equal. Therefore, we have proven the property of similarity between triangle δδ and triangle ε.ε.

    Your Turn 10.21

    1.
    Are these triangles similar? Find the lengths of sides /**/x/**/ and /**/y/**/ to prove your answer.
    Two right triangles. In the first triangle, the legs measure y and 18 meters. The hypotenuse measures 30 meters. In the second triangle, the legs measure 8 meters and x. The hypotenuse measures 10 meters. The top angles in both triangles are congruent.
    Figure 10.75

    Example 10.22

    Applying Similar Triangles

    A person who is 5 feet tall is standing 50 feet away from the base of a tree (Figure 10.76). The tree casts a 57-foot shadow. The person casts a 7-foot shadow. What is the height of the tree?

    An illustration of a palm tree and stick figure person shows a right triangle. The vertical leg shows a tree and it measures x. The hypotenuse is unknown. The horizontal leg measures 57 feet. A boy who is 5 feet tall is standing 50 feet away from the base of the tree. The boy casts a shadow of 7 feet.
    Figure 10.76
    Answer

    The bigger triangle includes a tree at side xx and the smaller triangle includes the person at the side labeled 5 ft. These two triangles are similar because the smaller triangle fits inside the larger triangle at the smallest angle. It would fit inside the larger triangle at either of the other two angles as well. That all angles are equal is one of the criteria for similar triangles, so we can solve using proportions:

    5 7 = x 57 5 ( 57 ) = 7 x 285 7 = x = 40.7 5 7 = x 57 5 ( 57 ) = 7 x 285 7 = x = 40.7

    The tree is 40.7 feet tall.

    Your Turn 10.22

    1.
    A person who is 6 feet tall is standing 100 feet away from the base of a tree. The tree casts a shadow 107.5-foot shadow. The person’s shadow is 7.5 feet long. How tall is the tree?
    An illustration of a palm tree and a stick figure person shows a right triangle. The vertical leg shows a tree. The hypotenuse is unknown. The horizontal leg measures 107.5 feet. A boy who is 6 feet tall is standing 100 feet away from the base of the tree. The boy casts a shadow of 7.5 feet.
    Figure 10.77

    Example 10.23

    Finding Missing Lengths

    At a certain time of day, a radio tower casts a shadow 180 feet long (Figure 10.78). At the same time, a 9-foot truck casts a shadow 15 feet long. What is the height of the tower?

    An illustration shows a right triangle. The vertical leg resembles a tower of x feet high. The horizontal leg measures 180 feet. A truck of 9 feet casts a shadow of 15 feet. The truck lies 15 feet to the left from the bottom-right of the triangle.
    Figure 10.78
    Answer

    These are similar triangles and the problem can be solved by using proportions:

    x 180 = 9 15 9 ( 180 ) = 15 x 1620 = 15 x 108 = x x 180 = 9 15 9 ( 180 ) = 15 x 1620 = 15 x 108 = x

    The height of the tower is 108 ft.

    Your Turn 10.23

    1.
    A tree casts a shadow of 180 feet early in the morning. A 10-foot high garage casts a shadow of 30 feet at the same time in the morning. What is the height of the tree?

    People in Mathematics

    Thales of Miletus

    Thales of Miletus, sixth century BC, is considered one of the greatest mathematicians and philosophers of all time. Thales is credited with being the first to discover that the two angles at the base of an isosceles triangle are equal, and that the two angles formed by intersecting lines are equal—that is, vertical or opposite angles, are equal. Thales is also known for devising a method for measuring the height of the pyramids by similar right triangles. Figure 10.79 shows his method. He measured the length of the shadow cast by the pyramid at the precise time when his own shadow ended at the same place.

    An illustration shows a right triangle. The vertical leg represents the height of the pyramid and it measures B prime. Sun is on the right. A vertical line, B is to the left of the pyramid. The horizontal leg measures A prime and the hypotenuse measures C prime. The distance from B to the bottom-left vertex along the horizontal leg is marked A. The distance from B to the bottom-left vertex along the hypotenuse is marked C.
    Figure 10.79 Thales and Similarity

    He equated the vertical height of the pyramid with his own height; the horizontal distance from the pyramid to the tip of its shadow with the distance from himself and the tip of his own shadow; and finally, the length of the shadow cast off the top of the pyramid with length of his own shadow cast off the top of his head. Using proportions, as shown in Figure 10.79, he essentially discovered the properties of similarity for right triangles. That is, ABCABC is similar to ABC.ABC. Note that to be similar, all corresponding angles between the two triangles must be equal, and the proportions from one side to another side within each triangle, as well as the proportions of the corresponding sides between the two triangles must be equal.

    Thales is also credited with discovering a method of determining the distance of a ship from the shoreline. Here is how he did it, as illustrated in Figure 10.80.

    Two triangles, A B C and C D E. The two triangles share the same point C. The second triangle is inverted. Point A is labeled ship. Point B is labeled shoreline. A B measures x. B C measures 60 feet. C D measures 45 feet. D E measures 330 feet.
    Figure 10.80 Thales and Similar Triangles

    Thales walked along the shoreline pointing a stick at the ship until it formed a 9090 angle to the shore. Then he walked along the shot and placed the stick in the ground at point CC. He continued walking until he reached point DD. Then, he turned and walked away from the shore at a 9090 angle until the stick he placed in the ground at point CC lined up with the ship, point EE. This is how he created similar triangles and estimated the distance of the ship to the shore by using proportions.

    Check Your Understanding

    17.
    Find the measure of the missing angle in the given figure.
    A triangle with its interior angles marked x, 26 degrees, and 65 degrees.
    18.
    Find the measure of the missing angle in the given figure.
    A right triangle with its interior angles marked x, 90 degrees, and 23 degrees.
    19.
    In the isosceles triangle shown, find the missing angles.
    A triangle with its interior angles marked x, y, and 77 degrees.
    20.
    In the figure shown given /**/\overline {BE}/**/ is parallel to /**/\overline {CD}/**/, find /**/x/**/ and /**/y/**/.
    A triangle, A C D with a horizontal line, B E at its center. A B measures 3. A E measures 4. B C measures 4. E D measures y. C D measures x. B E measures 2.
    21.
    Find /**/a/**/ and /**/t/**/ in the given figure.
    Two triangles, A B C and R S T. In the triangle A B C, the side A C measures a, the side B C measures 6, and the side A B measures 10. In the triangle R S T, the side R T measures 14, the side T S measures 12, and the side R S measures t. The angles, A and R are congruent. The angles, B and S are congruent.

    Section 10.3 Exercises

    For the following exercises, classify the triangle with the listed angle measurements as acute, right, or obtuse.
    1.
    /**/m\measuredangle A = {88^ \circ }/**/, /**/m\measuredangle B = {32^ \circ }/**/, /**/m\measuredangle C = {60^ \circ }/**/
    2.
    /**/m\measuredangle A = {65^ \circ }/**/, /**/m\measuredangle B = {90^ \circ }/**/, /**/m\measuredangle C = {25^ \circ }/**/
    3.
    /**/m\measuredangle A = {30^ \circ }/**/, /**/m\measuredangle B = {120^ \circ }/**/, /**/m\measuredangle C = {30^ \circ }/**/
    4.
    /**/m\measuredangle A = {85^ \circ }/**/, /**/m\measuredangle B = {50^ \circ }/**/, /**/m\measuredangle C = {45^ \circ }/**/
    5.
    Find the missing angles in the given figure.
    Two triangles, x y z, and a b c. The angles, x and a are congruent. The angle, x measures 74 degrees. The angles, y and b are congruent. The angle, b measures 28 degrees.
    6.
    What are the measurements of each angle in the given figure?
    A triangle, A B C. The angle A measures (4 x plus 7) degrees. The angle B measures (7 x plus 3) degrees. The angle C measures (23 x) degrees.
    7.
    Find the angle measurements in the given figure.
    A right triangle, A B C. Angle A measures 90 degrees. The angle B measures (4 x plus 2) degrees. The angle C measures (7 x plus 11) degrees.
    8.
    Find the angle measurements in the given figure.
    A triangle with its interior angles marked (x minus 2) degrees, (7 x minus 14) degrees, and (x minus 2) degrees.
    For the following exercises, determine which congruence theorem is used to show that the two triangles are congruent.
    9.
    Two triangles. The top angle of the first triangle and the top-right angle of the second triangle are congruent. The left side of the first triangle and the top side of the second triangle are congruent. The right sides of both triangles are congruent.
    10.
    Two right triangles. The hypotenuses of both triangles are congruent. The horizontal leg of the first triangle and the vertical leg of the second triangle are congruent.
    11.
    A kite, A B C D formed by joining two triangles. The sides, A D and D C are equal. The sides, A B and C B are equal.
    12.
    Are these triangles similar? If so, what is the common proportion or the scaling factor?
    Two right triangles. In the first triangle, the legs measure 1 and 2. In the second triangle, the legs measure 0.6 and 1.2. The top angles in both triangles are congruent.
    13.
    Are the images in the given figure similar? If so, what is the common proportion or the scaling factor?
    Two figures. Each figure shows two concentric squares. The inner square encloses a star. In the first figure, the outer square measures 4. The sides of the star measure 1. In the second figure, the outer square measures 6. The sides of the star measure 1.5.
    14.
    In the figure shown, given /**/\overline {BE}/**/ is parallel to /**/\overline {CD} ,/**/ find /**/x/**/ and /**/y/**/.
    A triangle, A C D with a horizontal line, B E at its center. A B measures 3. A E measures 4. B C measures 4. E D measures y. C D measures x. B E measures 2.
    15.
    Find /**/a/**/ and /**/t/**/ in the given figure.
    Two triangles, A B C and R S T. In the triangle A B C, the side A C measures a, the side B C measures 6, and the side A B measures 10. In the triangle R S T, the side R T measures 14, the side T S measures 12, and the side R S measures t. The angles, A and R are congruent. The angles, B and S are congruent.
    16.
    In the given figure, find the proportions and solve for /**/a/**/ and /**/b/**/.
    Two right triangles. In the first triangle, the legs measure a and b. The hypotenuse measures 25. In the second triangle, the legs measure 8 and 6. The hypotenuse measures 10.
    17.
    In the given figure, find the proportions and solve for /**/r/**/.
    Two triangles. In the first triangle, the top and right sides measure r and 21. In the second triangle, the top and right sides measure 12 and 4. The top-left angles in both triangles are congruent.
    18.
    A triangular plot of land has a perimeter of 2,400 ft. The longest side is 200 ft less than twice the shortest side. The middle side is 200 ft less than the longest side. Find the lengths of the three sides.
    19.
    The Orange Tree Hotel in Charleston, North Carolina has a fountain in the shape of a cylinder with a circular foundation. The circumference of the foundation is 6 times the radius increased by 12.88 ft. Find the radius of the circular foundation. (Use 3.14 as an approximation for /**/\pi/**/.)

    This page titled 10.4: Triangles is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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