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Chapter 2: Q21. (page 64)

Copy everything shown and write a two-column proof.

Given: AC¯⊥BC¯;∠3 is complementary to∠1.

Prove:∠3≅∠2

Short Answer

Expert verified

The two-column proof is:

Statement

Reason

AB¯⊥BC¯and∠3 is comp to∠1

Given

m∠3=m∠1=90°

∠3 is comp to∠1

m∠1+m∠2=90°

AB¯⊥BC¯

∠3+∠1=∠1=∠2

Both are equal to 90 degrees

m∠3=m∠2

Simplification

∠3≅∠2

Equal angles are congruent

Step by step solution

01

Step 1. Observe the given diagram.

The given diagram is:

02

Step 2. Write the relation between the angles ∠3 and ∠1.

As∠3 is complementary to∠1 , therefore the sum of the angles∠3 and∠1 is 90°.

Therefore, the relation between angles ∠3 and ∠1 is that the sum of angles ∠3 and ∠1 is 90°.

That implies, m∠3+m∠1=90°1.

03

Step 3. Write the relation between angles ∠1 and ∠2.

AsAC¯⊥BC¯, therefore the lines AC and BC are perpendicular, and therefore by using the definition of the perpendicular lines the measure of the angle ∠ACB=90°.

By using the angle addition postulate:

∠ACB=∠1+∠2.

Now,

∠ACB=90°∠1+∠2=90°2

04

Step 4. Write the relation between angles ∠3 and ∠2.

From the equations (1) and (2), it is obtained that:

m∠3+m∠1=m∠1+m∠2m∠3=m∠2

Therefore, the relation between the angles∠3 and∠2 is m∠3=m∠2.

That implies ∠3≅∠2.

05

Step 5. Write the two-column proof.

Steps:

Reason:

m∠3+m∠1=180

∠3 is supplementary to angle∠1

m∠4+m∠2=180

∠4 is supplementary to angle∠2

m∠1=m∠2

Vertical angles

m∠3+m∠1=m∠4+m∠2

Since both are equal to 180

m∠3=m∠4

Since,m∠1=m∠2

∠3≅∠4

Since, equal angles are congruent

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Most popular questions from this chapter

State which definition, postulate, or theorem justifies the statement about the diagram.

m∠1+m∠2=m∠BAC

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