Q18. Prove theorem 2-8: If two angles... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 2: Q18. (page 64)

Prove theorem 2-8: If two angles are complements of congruent angles, then the two angles are congruent. Note: You will need to draw your own diagram and state what is given and what you are to prove in terms of your diagram. (Hint: see the proof of theorem 2-7 on page 61.)

Short Answer

Expert verified

The diagram representing the complementary angles 鈭�1 and 鈭�2 and the complementary angles 鈭�3 and 鈭�4 is:

Given:鈭�1 and鈭�2 are complementary angles and鈭�3 and鈭�4 are complementary angles and $鈭� 1 鈮� 鈭� 3$ .

To Prove: $鈭� 2 鈮� 鈭� 4$ .

The two-column proof is:

Statements	Reasons
鈭�2 and鈭�1 are complementary. 鈭�3 and 鈭�4 are complementary.	Given
$m 鈭� 1 + m 鈭� 2 = 90 掳 m 鈭� 3 + m 鈭� 4 = 90 掳$	Definition of complementary angles
$m 鈭� 1 + m 鈭� 2 = m 鈭� 3 + m 鈭� 4$	Substitution property
$鈭� 1 鈮� 鈭� 3$ or $m 鈭� 1 = m 鈭� 3$	Given
$m 鈭� 2 = m 鈭� 4$ or $鈭� 2 鈮� 鈭� 4$	Subtraction property of equality.

Step by step solution

Step 1. Description of step.

Consider angles 鈭�1 and 鈭�2 as complementary angles and angles 鈭�3 and 鈭�4 as complementary angles.

Also consider angles 鈭�1 and 鈭�3 as congruent angles.

Therefore, given that 鈭�1 and 鈭�2 are complementary angles and 鈭�3 and 鈭�4 are complementary angles and $鈭� 1 鈮� 鈭� 3$

To prove that: $鈭� 2 鈮� 鈭� 4$

Step 2. Draw the diagram.

The diagram representing the complementary angles 鈭�1 and 鈭�2 and the complementary angles 鈭�3 and 鈭�4 is:

Step 3. Description of step.

It is being given that鈭�2 and鈭�1 are complementary and鈭�3 and鈭�4 are complementary.

Therefore, by using the definition of complementary angles it can be said that $m 鈭� 1 + m 鈭� 2 = 90 掳$ and $m 鈭� 3 + m 鈭� 4 = 90 掳$ .

As, $m 鈭� 1 + m 鈭� 2 = 90 掳$ and $m 鈭� 3 + m 鈭� 4 = 90 掳$ , therefore by using thesubstitution property it can be obtained that $m 鈭� 1 + m 鈭� 2 = m 鈭� 3 + m 鈭� 4$ .

It is being given that $鈭� 1 鈮� 鈭� 3$ or $m 鈭� 1 = m 鈭� 3$ .

Therefore, it can be obtained that:

$\begin{matrix} m 鈭� 1 + m 鈭� 2 = m 鈭� 3 + m 鈭� 4 \\ m 鈭� 1 + m 鈭� 2 = m 鈭� 1 + m 鈭� 4 \\ m 鈭� 2 = m 鈭� 4 \end{matrix}$

Therefore, $鈭� 2 鈮� 鈭� 4$ .

Hence proved.

Step 4. Description of step.

Statements	Reasons
鈭�2 and鈭�1 are complementary. 鈭�3 and鈭�4 are complementary.	Given
$m 鈭� 1 + m 鈭� 2 = 90 掳 m 鈭� 3 + m 鈭� 4 = 90 掳$	Definition of complementary angles
$m 鈭� 1 + m 鈭� 2 = m 鈭� 3 + m 鈭� 4$	Substitution property
$鈭� 1 鈮� 鈭� 3$ or $m 鈭� 1 = m 鈭� 3$	Given
$m 鈭� 2 = m 鈭� 4$ or $鈭� 2 鈮� 鈭� 4$	Subtraction property of equality.