Question

1. in the figure below, line m and line n are parallel. the 2 transversals intersect on a point on line m. which of the following pairs of angles is not necessarily congruent? a. ∠1 and ∠2 b. ∠2 and ∠3 c. ∠2 and ∠4 d. ∠3 and ∠4 e. ∠3 and ∠5 2. in the diagram below, lines l and k are cut by transversal r. lines l and k are not parallel. which of the following must be true? f. ∠1 ≅ ∠5 g. ∠2 ≅ ∠8 h. ∠5 ≅ ∠6 j. ∠5 ≅ ∠7 k. ∠7 ≅ ∠3 3. the sum of ∠b and ∠c is 90°, and ∠a ≅ ∠c. what must be true about the measures of ∠a and ∠b? a. m∠a + m∠b = 90° b. m∠a + m∠b = 180° c. m∠b > m∠a d. m∠b < m∠a e. m∠b = m∠a

Explanation:

Problem 1

Step1: Analyze each angle pair

• A. $\angle1$ and $\angle2$: Supplementary (linear pair), not congruent unless right angles, but wait no—wait, no, $\angle1$ and $\angle2$ are adjacent on a line, so they are supplementary, but wait no, the question asks which is NOT necessarily congruent. Wait, recheck:
• A. $\angle1$ and $\angle2$: Linear pair, sum to $180^\circ$, only congruent if both $90^\circ$, so not necessarily congruent? No, wait no, let's check others:
• B. $\angle2$ and $\angle3$: Alternate interior angles (parallel lines $m\parallel n$), so congruent.
• C. $\angle2$ and $\angle4$: $\angle2\cong\angle3$ (alt int), $\angle3\cong\angle4$ (vertical angles), so $\angle2\cong\angle4$.
• D. $\angle3$ and $\angle4$: Vertical angles, congruent.
• E. $\angle3$ and $\angle5$: Vertical angles, congruent.

Wait, no, A: $\angle1$ and $\angle2$ are linear pair, so they are supplementary, not congruent unless perpendicular transversal. So A is not necessarily congruent.

Problem 2

Step1: Analyze each angle pair

• F. $\angle1\cong\angle5$: Corresponding angles, only congruent if lines are parallel (lines $l,k$ are NOT parallel), so not true.
• G. $\angle2\cong\angle8$: Alternate exterior angles, only congruent if parallel, not true.
• H. $\angle5\cong\angle6$: Linear pair, supplementary, not congruent unless right angles, not true.
• J. $\angle5\cong\angle7$: Vertical angles, always congruent, regardless of parallel lines.
• K. $\angle7\cong\angle3$: Corresponding angles, only congruent if parallel, not true.

Problem 3

Step1: Substitute congruent angle

Given $m\angle B + m\angle C = 90^\circ$, and $\angle A\cong\angle C$, so $m\angle A = m\angle C$.

Step2: Replace $\angle C$ with $\angle A$

Substitute into the sum: $m\angle B + m\angle A = 90^\circ$.

Answer:

1. A. $\angle1$ and $\angle2$
2. J. $\angle5 \cong \angle7$
3. A. $\text{m}\angle A + \text{m}\angle B = 90^\circ$