# 4.E: Basic Concepts of Euclidean Geometry (Exercises)

- Page ID
- 4898

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

## Exercise \(\PageIndex{1}\): Similarity

Assess whether each of the following statement is true or false and justify your answer.

- Any two equilateral triangles are similar.
- Any two isosceles triangles are similar.
- Any two squares are similar.
- Any two rectangles are similar.
- Any two pentagons are similar.
- Any two polygons are similar.

## Exercise \(\PageIndex{2}\): Area

Starting with the formula for the area of a triangle, show how to obtain the formula for the area of a parallelogram and the area of a hexagon.

## Exercise \(\PageIndex{3}\): Interior angle

Find the general formula for the center angle in a regular n-sided polygon? Justify your answer.

Find the general formula for the exterior angle in a regular n-sided polygon? Justify your answer.

## Exercise \(\PageIndex{4}\): Triangles

Prove or disprove the following: an equilateral triangle is an isosceles triangle.

explain why an equilateral triangle is not a scalene triangle?

## Exercise \(\PageIndex{5}\): True or False

Explain why the following statements below are true.

- A square is a rectangle.
- A rectangle is a parallelogram.
- A square is a kite.
- A parallelogram is a trapezoid.

## Exercise \(\PageIndex{6}\): Angles

Given cutout sheets with angles \(40^{\circ}, 55^{\circ }\) and \(85^{\circ}\).** **By adding or subtracting angles, construct other angles that measure \(15^{\circ}, 30^{\circ}, 70^{\circ}, 95^{\circ} \)and \(100^{\circ}.\)

Can you construct an angle that measures \( 75^{\circ}? \)Explain how or say why not.

## Exercise \(\PageIndex{7}\): Converse of the Pythagorean Theorem

State the statement for the converse of the Pythagorean Theorem. Is this statement true or false? Justify your answer.

## Exercise \(\PageIndex{8}\): Venn diagram

Create a Venn diagram to illustrate the types of quadrilaterals, listed in this section.

## Exercise \(\PageIndex{9}\): Parallelogram

Let \((a; b); ((0; c); (d:e); (f; 0)\) be vertices of a quadrilateral. Show that if you take the midpoints of any quadrilateral and connect them in turn, you will always get a parallelogram.

## Exercise \(\PageIndex{10}\): Venn Triangles

Express the relationship between scalene, isosceles and equilateral triangles with Venn diagram.

## Exercise \(\PageIndex{11}\): Angles

Consider the figure:

Given that \(a= 47°,\) and \(c= 55°.\)

Find other listed angles.

## Exercise \(\PageIndex{12}\):

Refer to the cube picture above.

Name the following:

## Exercise \(\PageIndex{13}\)

Find the area between the perimeter of this square and the unit circle.

**Answer**-
\(1- \pi\)

## Exercise \(\PageIndex{14}\)

Find the area between the perimeter of the unit circle and the triangle created from connecting the points \((0,1), (−\frac{4}{5}, −\frac{3}{5}) \)and \(\displaystyle (\frac{4}{5}, −\frac{3}{5}) \), as seen in the following figure.

**Answer**-
(\displaystyle π−\frac{32}{25}\)

By Illuyanka (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

Exercise 13 and 14 are from

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.