Question

1. in the figure below, bd is parallel to eg. which of the following pairs of angles must be congruent to each other? a. ∠acb,∠acd b. ∠acb,∠cfg c. ∠acb,∠bcf d. ∠acd,∠cfe e. ∠acd,∠cfg 10. which of the following is equal to the expression 3(a + 1)+5 when a = -7? f. -23 g. -18 h. -13 j. 23 k. 29

Explanation:

9.

Step1: Recall parallel - line angle properties

When two parallel lines ($BD$ and $EG$) are cut by a transversal ($AF$), corresponding angles are congruent, alternate - interior angles are congruent, and alternate - exterior angles are congruent.

Step2: Analyze each option

• Option A: $\angle ACB$ and $\angle ACD$ are a linear pair, so $\angle ACB+\angle ACD = 180^{\circ}$, not congruent.
• Option B: $\angle ACB$ and $\angle CFG$ are not corresponding, alternate - interior, or alternate - exterior angles.
• Option C: $\angle ACB$ and $\angle BCF$ are a linear pair, so $\angle ACB+\angle BCF = 180^{\circ}$, not congruent.
• Option D: $\angle ACD$ and $\angle CFE$ are alternate - interior angles. When two parallel lines $BD$ and $EG$ are cut by a transversal $AF$, alternate - interior angles are congruent.
• Option E: $\angle ACD$ and $\angle CFG$ are not corresponding, alternate - interior, or alternate - exterior angles.

Step1: Simplify the expression

First, simplify the expression $3(a + 1)+5$. Using the distributive property, $3(a + 1)+5=3a+3 + 5=3a + 8$.

Step2: Substitute $a=-7$

Substitute $a = - 7$ into the simplified expression $3a + 8$. Then $3(-7)+8=-21 + 8=-13$.

Answer:

D. $\angle ACD,\angle CFE$

10.