1.4.2: Adding the Angles in a Triangle

Lesson

Let's explore angles in triangles.

Exercise \(\PageIndex{1}\): Can You Draw It?

1. Complete the table by drawing a triangle in each cell that has the properties listed for its column and row. If you think you cannot draw a triangle with those properties, write “impossible” in the cell.
2. Share your drawings with a partner. Discuss your thinking. If you disagree, work to reach an agreement.

Table \(\PageIndex{1}\)

	acute (all angles acute)	right (has a right angle)	obtuse (has an obtuse angle)
scalene (side lengths all different)
isosceles (at least two side lengths are equal)
equilateral (three side lengths equal)

Exercise \(\PageIndex{2}\): Find All Three

Your teacher will give you a card with a picture of a triangle.

1. The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.
2. Find two other students with triangles congruent to yours but with a different angle labeled. Confirm that the triangles are congruent, that each card has a different angle labeled, and that the angle measures make sense.
3. Enter the three angle measures for your triangle on the table your teacher has posted.

Exercise \(\PageIndex{3}\): Tear It Up

Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?

Are you ready for more?

1. Draw a quadrilateral. Cut it out, tear off its angles, and line them up. What do you notice?
2. Repeat this for several more quadrilaterals. Do you have a conjecture about the angles?

Summary

A \(180^{\circ}\) angle is called a straight angle because when it is made with two rays, they point in opposite directions and form a straight line.

Figure \(\PageIndex{1}\)

If we experiment with angles in a triangle, we find that the sum of the measures of the three angles in each triangle is \(180^{\circ}\)—the same as a straight angle!

Through experimentation we find:

• If we add the three angles of a triangle physically by cutting them off and lining up the vertices and sides, then the three angles form a straight angle.
• If we have a line and two rays that form three angles added to make a straight angle, then there is a triangle with these three angles.

Figure \(\PageIndex{2}\)

Glossary Entries

Definition: Alternate Interior Angles

Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

This diagram shows two pairs of alternate interior angles. Angles \(a\) and \(d\) are one pair and angles \(b\) and \(c\) are another pair.

Figure \(\PageIndex{3}\): There are two horizontal parallel lines, and a third diagonal line drawn from the bottom left to the upper right, intersecting both horizontal lines. The diagonal line is labeled transversal. There are four angles created by the diagonal line inside the parallel lines. The upper left angle is labeled a, upper right is b, lower left is c, and lower right is d.

Definition: Straight Angle

A straight angle is an angle that forms a straight line. It measures 180 degrees.

Figure \(\PageIndex{4}\)

Definition: Transversal

A transversal is a line that crosses parallel lines.

This diagram shows a transversal line \(k\) intersecting parallel lines \(m\) and \(l\).

Figure \(\PageIndex{5}\)

Practice

Exercise \(\PageIndex{4}\)

In triangle \(ABC\), the measure of angle \(A\) is \(40^{\circ}\).

1. Give possible measures for angles \(B\) and \(C\) if triangle \(ABC\) is isosceles.
2. Give possible measures for angles \(B\) and \(C\) if triangle \(ABC\) is right.

Exercise \(\PageIndex{5}\)

For each set of angles, decide if there is a triangle whose angles have these measures in degrees:

1. 60, 60, 60
2. 90, 90, 45
3. 30, 40, 50
4. 90, 45, 45
5. 120, 30, 30

If you get stuck, consider making a line segment. Then use a protractor to measure angles with the first two angle measures.

Exercise \(\PageIndex{6}\)

Angle \(A\) in triangle \(ABC\) is obtuse. Can angle \(B\) or angle \(C\) be obtuse? Explain your reasoning.

Exercise \(\PageIndex{7}\)

For each pair of polygons, describe the transformation that could be applied to Polygon A to get Polygon B.

1.

Figure \(\PageIndex{6}\)

2.

Figure \(\PageIndex{7}\)

3.

Figure \(\PageIndex{8}\)

(From Unit 1.1.3)

Exercise \(\PageIndex{8}\)

On the grid, draw a scaled copy of quadrilateral \(ABCD\) using a scale factor of \(\frac{1}{2}\).

Figure \(\PageIndex{9}\): A quadrilateral on a grid. Point A is 2 units right and 6 units up. Point B is 4 units right and 8 units up. Point C is 8 units right and 6 units up. Point D is 4 units right and 2 units up.

(From Unit 1.4.1)