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Chapter 2: Q9. (page 62)

Describe your plan for proving the following. You don’t need to give all the details.

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

Prove: ∠3≅∠2

Short Answer

Expert verified

It is proven that∠3≅∠2.

Step by step solution

01

Step 1. Observe the given diagram.

The given diagram is:

02

Step 2. Definition of complementary angles.

The complementary angles are the angles whose sum is 90°.

03

Step 3. Write the proof of the statement ∠3≅∠2.

∠3 is complementary to∠1.

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

That implies,

∠3+∠1=90°1

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

Subtract the equation (2) from (1).

∠3+∠1−∠1+∠2=90°−90°∠3+∠1−∠1−∠2=0∠3−∠2=0∠3=∠2

Therefore, ∠3≅∠2.

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