Question

activity 6 practice
write your answers on notebook paper.
show your work.
lesson 6 - 1
use this diagram to identify the property, postulate,
or theorem that justifies each statement in items 1 - 4.

1. ( pq+qr = pr )

a. angle addition postulate
b. addition property
c. definition of congruent segments
d. segment addition postulate

1. if ( q ) is the midpoint of ( overline{pr} ), then ( overline{pq}congoverline{qr} ).

a. definition of midpoint
b. definition of congruent segments
c. definition of segment bisector
d. segment addition postulate

1. ( angle 3congangle 4 )

a. definition of supplementary
b. definition of congruent angles
c. vertical angles are congruent
d. definition of angle bisector

1. if ( angle 1 ) is complementary to ( angle 2 ), then ( mangle 1+mangle 2 = 90^{circ} ).

a. angle addition postulate
b. addition property
c. definition of perpendicular
d. definition of complementary

Explanation:

Question 1

Step 1: Recall the Segment Addition Postulate

The Segment Addition Postulate states that if we have three points \( P \), \( Q \), and \( R \) that are collinear (lie on the same line) with \( Q \) between \( P \) and \( R \), then \( PQ + QR = PR \).

Step 2: Match with the options

• Option A is about angle addition, not segment, so incorrect.
• Option B is the Addition Property (like for equations), not for segments, incorrect.
• Option C is about congruent segments (defining what congruent segments are), not about adding segments, incorrect.
• Option D is the Segment Addition Postulate, which matches the statement \( PQ + QR = PR \).

Step 1: Recall the Definition of Midpoint

The definition of a midpoint of a segment \( PR \) is a point \( Q \) that divides \( PR \) into two equal (congruent) segments, so \( PQ \cong QR \) (and \( PQ = QR \)).

Step 2: Match with the options

• Option A: Definition of midpoint says that the midpoint divides the segment into two congruent segments, so this matches.
• Option B: Definition of congruent segments is about what makes two segments congruent (same length), not about midpoint, incorrect.
• Option C: Definition of segment bisector is a line, ray, or segment that divides another segment into two equal parts (it doesn't have to be a midpoint, could be a bisector not starting from an endpoint), so incorrect.
• Option D: Segment Addition Postulate is about adding segments, not about midpoint defining congruent segments, incorrect.

Step 1: Recall Vertical Angles Theorem

Vertical angles are the angles opposite each other when two lines intersect. The Vertical Angles Theorem states that vertical angles are congruent (have equal measure). Here, \( \angle 3 \) and \( \angle 4 \) are vertical angles (formed by the intersection of lines \( AE \) and \( PR \)).

Step 2: Match with the options

• Option A: Definition of supplementary angles is about two angles adding to \( 180^\circ \), not about vertical angles, incorrect.
• Option B: Definition of congruent angles is just saying what congruent angles are (equal measure), not why \( \angle 3 \) and \( \angle 4 \) are congruent, incorrect.
• Option C: Vertical angles are congruent (the theorem), which matches \( \angle 3 \cong \angle 4 \), correct.
• Option D: Definition of angle bisector is about dividing an angle into two equal angles, not about vertical angles, incorrect.

Answer:

D. Segment Addition Postulate

Question 2