Question

grade 10 geometry
district formative assessment – extended response
name
date
er/dfa1.geo.m.g.co.09. prove theorems about lines and angles. theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

1. choose the theorem that would best be used to prove the following statement: given that lines a and b are parallel, if the m∠3 = 115°, then m∠5 = 65°.

a) alternate interior angles theorem
b) alternate exterior angles theorem
c) consecutive interior angles theorem
d) vertical angles theorem

1. which statement proves ∠3 ≅ ∠5?

a) alternate interior angles theorem
b) alternate exterior angles theorem
c) corresponding angles postulate
d) vertical angles theorem
(diagram with lines m, n, transversal p, angles 1–8)

1. given the following figure, solve for y.

(diagram with two parallel lines, transversal, angle 125°, angle (7y + 27)°)
a) y = 4
b) y = 14
c) y = 22
d) y = 34
revised 6/15/22
© vail school district 2013

Explanation:

Question 1

Step1: Analyze angle relationships

We know that \( m\angle3 = 115^\circ \) and \( m\angle5 = 65^\circ \). Check if they are supplementary (sum to \( 180^\circ \)): \( 115^\circ+ 65^\circ=180^\circ \).

Step2: Identify the theorem

Consecutive Interior Angles Theorem states that if two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Since \( \angle3 \) and \( \angle5 \) are consecutive interior angles and supplementary, the theorem is Consecutive Interior Angles Theorem.

Step1: Recall angle theorems for congruence

To prove \( \angle3\cong\angle5 \), we check the relationship. \( \angle3 \) and \( \angle5 \) are alternate interior angles (formed by transversal \( P \) cutting parallel lines \( m \) and \( n \)).

Step2: Identify the theorem

Alternate Interior Angles Theorem states that alternate interior angles are congruent when lines are parallel.

Step1: Identify angle relationship

The two angles \( (7y + 27)^\circ \) and \( 125^\circ \) are same - side interior angles (or supplementary as lines are parallel, so they should be supplementary? Wait, no, looking at the diagram, they are actually same - side interior angles? Wait, no, if the lines are parallel, then \( (7y + 27)^\circ+ 125^\circ = 180^\circ \)? Wait, no, wait the angle given is \( 125^\circ \) and the other angle is \( (7y + 27)^\circ \). Wait, maybe they are supplementary? Wait, no, let's re - examine. Wait, if the two lines are parallel, and the transversal cuts them, then the angle \( (7y + 27)^\circ \) and \( 125^\circ \) are supplementary? Wait, no, maybe they are equal? Wait, no, the diagram shows that one angle is \( 125^\circ \) and the other is \( (7y + 27)^\circ \). Wait, maybe they are supplementary. So:
\( 7y+27 + 125=180 \)

Step2: Solve for y

First, combine like terms: \( 7y+152 = 180 \)
Subtract 152 from both sides: \( 7y=180 - 152=28 \)
Divide both sides by 7: \( y = \frac{28}{7}=4 \)? Wait, no, that's not right. Wait, maybe the angle \( (7y + 27)^\circ \) is equal to \( 125^\circ \)? Wait, no, maybe I made a mistake. Wait, let's check the options. The options are \( y = 4,y = 14,y = 22,y = 34 \). Wait, maybe the angle \( (7y + 27)^\circ \) and \( 125^\circ \) are supplementary? Wait, \( 7y+27+125 = 180\Rightarrow7y=28\Rightarrow y = 4 \), but option a is \( y = 4 \). Wait, maybe the angle \( (7y + 27)^\circ \) is equal to \( 125^\circ \)? No, \( 7y+27 = 125\Rightarrow7y=98\Rightarrow y = 14 \). Ah, maybe I misidentified the angle relationship. If the lines are parallel, and the angles are alternate interior angles or corresponding angles, then they are equal. So if \( 7y + 27=125 \)

Step2: Solve for y

Subtract 27 from both sides: \( 7y=125 - 27 = 98 \)
Divide by 7: \( y=\frac{98}{7}=14 \)

Answer:

c) Consecutive Interior Angles Theorem

Question 2