Question

Angie and Becky each completed a separate proof to show that the measures of vertical angles AKG and HKB are equal. Who completed the proof incorrectly? Explain.
Angie's:

Statement Justification
1. Segment GH intersects segment AB at K 1. Given
2. m∠AKG + m∠GKB = 180°
m∠GKB + m∠HKB = 180° 2. Definition of Supplementary Angles
3. m∠AKG + m∠ GKB = m∠GKB + m∠HKB 3. Substitution Property
4. m∠AKG = m∠HKB 4. Subtraction Property

Becky's:

Statement Justification
1. Segment GH intersects segment AB at K 1. Given
2. m∠AKG + m∠GKB = 180°
m∠GKB + m∠HKB = 180° 2. Angle Addition Postulate
3. m∠AKG + m∠ GKB = m∠GKB + m∠HKB 3. Substitution Property
4. m∠AKG = m∠HKB 4. Subtraction Property

Answer 1

Vertical angles are angles which are opposite to each other and have equal measures. With respect to the given proof, Angie completed the proof incorrectly.

Two angles are said to be vertical when they are opposite to one another and have equal measures. However, two angles are supplementary if they add up to 180°.

So, comparing Angie's and Becky's proofs, it can be deduced that from Angies proof:

1. Segment GH intersects segment AB at K (1. Given)

2. m∠AKG + m∠GKB = 180° (2. Definition of Supplementary Angles)

   m∠GKB + m∠HKB = 180°   (2. Definition of Supplementary Angles)

3. m∠AKG + m∠ GKB = m∠GKB + m∠HKB  (3. Substitution Property)

4. m∠AKG = m∠HKB (4. Subtraction Property)

Note:

• Step 2 should be Angle Addition Postulate, since:

   m∠AKG + m∠GKB = 180°

   m∠GKB + m∠HKB = 180°

So that;

   m∠AKG + m∠GKB = m∠GKB + m∠HKB

Subtract m∠GKB from both sides to have;

m∠AKG = m∠HKB (which is the proof for step 4)

Therefore, Angie completed the poof incorrectly because of the properties of steps 2 and 4.

A diagram is herewith attached for more clarifications.

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