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Chapter 2: Q15. (page 63)

Copy everything shown and complete the proof of theorem 2-7 for the case where two angles are supplements of the same angle.

Given:∠1 and∠5 are supplementary;∠3 and∠5 are supplementary.

Prove: ∠1≅∠3

Short Answer

Expert verified

The complete proof table is:

Statements

Reasons

∠1 and∠5 are supplementary.

∠3 and∠5 are supplementary.

GIven

m∠1+m∠5=180;m∠3+m∠5=180


As,∠1 and∠5 are supplementary angles and

∠3 and ∠5 are supplementary angles

m∠1+m∠5=m∠3+m∠5

Substitution property

m∠5=m∠5

Reflexive property

m∠1=m∠3or∠1≅∠3

Solve the equation m∠1+m∠5=m∠3+m∠5.

Step by step solution

01

Step 1. Observe the given diagram.

The given diagram is:

02

Step 2. Description of step.

It is being given that∠1 and∠5 are supplementary and∠3 and∠5 are supplementary.

As, and are supplementary angles, therefore the sum of the angles∠1 and∠5 is 180°.

Therefore, m∠1+m∠5=180.

As,∠3 and∠5 are supplementary angles, therefore the sum of the angles∠3 and∠5 is 180°.

Therefore, m∠3+m∠5=180.

Now, by using the substitution property it can be obtained that:

m∠1+m∠5=m∠3+m∠5

Therefore, it can be obtained that:

m∠1+m∠5=m∠3+m∠5m∠1=m∠3⇒∠1≅∠3

Hence proved.

03

Step 3. Complete the proof table.

Statements

Reasons

∠1 and∠5 are supplementary.

∠3 and∠5 are supplementary.

GIven

m∠1+m∠5=180;m∠3+m∠5=180


As,∠1 and∠5 are supplementary angles and

∠3 and∠5 are supplementary angles

m∠1+m∠5=m∠3+m∠5

Substitution property

m∠5=m∠5

Reflexive property

m∠1=m∠3or∠1≅∠3

Solve the equation m∠1+m∠5=m∠3+m∠5.

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