Question

complete the proof.
given:
m∠1 + m∠2 = 90
prove: m∠3 = 90
diagram: vertical and horizontal lines with angles 1, 2, 3; below diagram: boxes with ∠1 ≅ ∠3, m∠1 + m∠2 = m∠3, the sum of ∠1 and ∠2, m∠3 = 90, ∠1
∠3 forms a vertical angle pair with __. so because vertical angles are congruent. since m∠1 + m∠2 = 90 is given, __ by the transitive property of equality

Explanation:

Step1: Identify vertical angle pair

∠3 and ∠1 form a vertical angle pair (from the diagram, vertical angles are opposite each other when two lines intersect). So the first blank is ∠1.

Step2: Vertical angles congruence

Since vertical angles are congruent, \( \angle 1 \cong \angle 3 \), which implies \( m\angle 1 = m\angle 3 \) (or in the given options, \( \angle 1 \cong \angle 3 \) is the relevant statement here, but the next part: from vertical angles congruent, we can say \( m\angle 1 + m\angle 2 = m\angle 3 \)? Wait, no, let's re - examine. Wait, ∠1 and ∠3 are vertical angles, so \( \angle 1\cong\angle 3 \), so \( m\angle 1=m\angle 3 \). But also, from the diagram, ∠1, ∠2 and the right angle? Wait, no, the given is \( m\angle 1 + m\angle 2=90 \), and we need to prove \( m\angle 3 = 90 \).

Wait, ∠3 forms a vertical angle pair with ∠1 (the angle between the vertical line and the horizontal line? Wait, looking at the diagram, the vertical line and horizontal line intersect, so ∠3 and the angle adjacent to ∠1 and ∠2? Wait, maybe the first blank: ∠3 forms a vertical angle pair with ∠1 (the right angle? No, wait, the diagram has a vertical line, a horizontal line, and a ray making ∠1 and ∠2 with the vertical and horizontal lines. So ∠3 is vertical to the angle that is \( \angle 1+\angle 2 \)? Wait, no, let's use the given options.

The first blank: ∠3 forms a vertical angle pair with ∠1? Wait, no, maybe the angle that is \( \angle 1+\angle 2 \) and ∠3? Wait, the options are \( \angle 1\cong\angle 3 \), \( m\angle 1 + m\angle 2=m\angle 3 \), "the sum of ∠1 and ∠2", \( m\angle 3 = 90 \), \( \angle 1 \).

First blank: ∠3 forms a vertical angle pair with the angle that is \( \angle 1+\angle 2 \)? No, the first blank is a single angle. Wait, the vertical angle of ∠3: in the diagram, the vertical line and horizontal line intersect, so ∠3 and the angle opposite to it (let's say the angle at the top left) but in the given, we have ∠1, ∠2, ∠3. Wait, maybe ∠3 is vertical to the angle that is \( \angle 1+\angle 2 \)? No, the first blank is to fill with an angle. The options have \( \angle 1 \). So ∠3 forms a vertical angle pair with \( \angle 1 \)? Wait, no, maybe the angle that is \( \angle 1+\angle 2 \) is equal to ∠3? Wait, let's think again.

Given \( m\angle 1 + m\angle 2 = 90 \), we need to prove \( m\angle 3=90 \). So ∠3 and the angle \( \angle 1+\angle 2 \) are vertical angles? Wait, the diagram: the vertical line and horizontal line are perpendicular? Wait, no, the horizontal line and vertical line intersect, so the angle between them is 90 degrees. Wait, ∠3 is adjacent to the horizontal line and vertical line, so ∠3 is 90 degrees? No, the proof is to show that.

Wait, the first blank: ∠3 forms a vertical angle pair with the angle that is \( \angle 1+\angle 2 \)? No, the first blank is a single angle. The options have \( \angle 1 \). So ∠3 forms a vertical angle pair with \( \angle 1 \)? No, maybe the angle composed of ∠1 and ∠2. Wait, the first blank: "∠3 forms a vertical angle pair with" then the answer is \( \angle 1 \)? No, let's use the options.

The first blank: the angle that ∠3 is vertical to. From the options, the first option is \( \angle 1\cong\angle 3 \), the second is \( m\angle 1 + m\angle 2=m\angle 3 \), the third is "the sum of ∠1 and ∠2", the fourth is \( m\angle 3 = 90 \), the fifth is \( \angle 1 \).

So first blank: ∠3 forms a vertical angle pair with \( \angle 1 \)? No, maybe the angle that is \( \angle 1+\angle 2 \) is equal to ∠3. Wait, the second part: "So" then the reason is vertical angles are congruent, so the statement should be \( \angle 1\cong\angle 3 \) or \( m\angle 1 + m\angle 2=m\angle 3 \). Wait, if ∠3 is vertical to the angle \( \angle 1+\angle 2 \), then \( m\angle 1 + m\angle 2=m\angle 3 \) because vertical angles are congruent. Then, since \( m\angle 1 + m\angle 2 = 90 \) (given), by transitive property, \( m\angle 3=90 \).

So let's re - structure:

1. ∠3 forms a vertical angle pair with the angle that is the sum of ∠1 and ∠2? No, the first blank is an angle. Wait, the diagram: the vertical line, horizontal line, and a ray. So ∠1 is between vertical and ray, ∠2 between ray and horizontal, ∠3 between horizontal and vertical (below). So the angle above (opposite to ∠3) is \( \angle 1+\angle 2 \). So ∠3 and \( \angle 1+\angle 2 \) are vertical angles. So first blank: "the sum of ∠1 and ∠2"? No, the first blank is to fill with an angle. Wait, the options have "∠1", "∠1≅∠3", "m∠1 + m∠2 = m∠3", "the sum of ∠1 and ∠2", "m∠3 = 90".

Wait, the first blank: ∠3 forms a vertical angle pair with ∠1? No, let's look at the sentence structure: "∠3 forms a vertical angle pair with [blank]. So [blank] because vertical angles are congruent. Since \( m\angle 1 + m\angle 2 = 90 \) is given, [blank] by the transitive property of equality".

First blank: the angle that ∠3 is vertical to. From the diagram, ∠3 is vertical to the angle that is \( \angle 1+\angle 2 \)? No, the first blank is a single angle. Wait, maybe the angle at the top (let's say ∠A) which is equal to \( \angle 1+\angle 2 \), but the options don't have that. Wait, the options have "∠1", so maybe ∠3 is vertical to ∠1? No, that doesn't make sense. Wait, maybe the problem has a typo, but let's use the options.

First blank: ∠1 (option 5). Then "So \( \angle 1\cong\angle 3 \)" (option 1) because vertical angles are congruent. Then, since \( m\angle 1 + m\angle 2 = 90 \), and \( m\angle 1=m\angle 3 \) (from \( \angle 1\cong\angle 3 \)), then \( m\angle 3 + m\angle 2=90 \)? No, that's not helpful.

Wait, another approach: the angle \( \angle 1+\angle 2 \) and ∠3 are vertical angles. So first blank: "the sum of ∠1 and ∠2" (option 3). Then "So \( m\angle 1 + m\angle 2=m\angle 3 \)" (option 2) because vertical angles are congruent (their measures are equal). Then, since \( m\angle 1 + m\angle 2 = 90 \) (given), by transitive property, \( m\angle 3 = 90 \) (option 4).

Yes, that makes sense. So:

- ∠3 forms a vertical angle pair with the sum of ∠1 and ∠2 (so first blank: "the sum of ∠1 and ∠2").
- So \( m\angle 1 + m\angle 2=m\angle 3 \) (second blank: \( m\angle 1 + m\angle 2=m\angle 3 \)) because vertical angles are congruent (their measures are equal).
- Since \( m\angle 1 + m\angle 2 = 90 \) is given, \( m\angle 3 = 90 \) (third blank: \( m\angle 3 = 90 \)) by the transitive property of equality (if \( a = b \) and \( b = c \), then \( a = c \); here \( a=m\angle 3 \), \( b=m\angle 1 + m\angle 2 \), \( c = 90 \)).

Answer:

First blank: the sum of ∠1 and ∠2
Second blank: \( m\angle 1 + m\angle 2=m\angle 3 \)
Third blank: \( m\angle 3 = 90 \)