Question

use the diagram at the right. complete each statement. 9. ∠mig≅____ 10. ∠pmj≅__ 11. if m∠kjl=30, then m∠__=30. 12. if m∠lmp=100, then m∠qhg=____.

Explanation:

Question 9

Step1: Identify Vertical Angles

Vertical angles are equal. $\angle MIG$ and $\angle LIH$ are vertical angles (formed by intersecting lines).

Step2: Conclude Congruence

Since vertical angles are congruent, $\angle MIG \cong \angle LIH$.

Step1: Identify Vertical Angles

$\angle PMJ$ and $\angle QMI$ are vertical angles (formed by intersecting lines $PQ$ and the other line through $M$).

Step2: Conclude Congruence

Vertical angles are congruent, so $\angle PMJ \cong \angle QMI$.

Step1: Identify Corresponding or Vertical Angles

$\angle KJL$ and $\angle GIM$ (or $\angle MIG$? Wait, no—wait, $\angle KJL$ and $\angle GIM$? Wait, actually, $\angle KJL$ and $\angle GIM$? Wait, no, looking at the diagram, $\angle KJL$ and $\angle GIM$? Wait, no, $\angle KJL$ and $\angle MIG$? Wait, no, let's see: lines $KL$ and $GI$ are parallel? Wait, maybe corresponding angles or vertical. Wait, $\angle KJL$ and $\angle MIG$? No, wait, $\angle KJL$ and $\angle GIM$? Wait, actually, $\angle KJL$ and $\angle MIG$? Wait, no, the correct angle: $\angle KJL$ and $\angle GIM$? Wait, no, let's think again. $\angle KJL$ and $\angle MIG$? Wait, no, the vertical or corresponding. Wait, $\angle KJL$ and $\angle GIM$? Wait, maybe $\angle MIG$? No, wait, the answer is $\angle MIG$? Wait, no, let's check. If $\angle KJL = 30^\circ$, then the angle congruent to it (vertical or corresponding) would be $\angle MIG$? Wait, no, maybe $\angle GIM$? Wait, no, the correct angle is $\angle MIG$? Wait, no, let's see the diagram. $\angle KJL$ and $\angle MIG$: are they vertical angles? Wait, lines $KL$ and $GI$ intersect at $J$ and $I$? Wait, maybe $\angle KJL$ and $\angle GIM$ are corresponding angles. Wait, maybe the answer is $\angle MIG$? Wait, no, let's do it properly. $\angle KJL$ and $\angle MIG$: are they equal? Wait, maybe $\angle KJL \cong \angle MIG$ because they are corresponding angles (if lines $KL$ and $GI$ are parallel, and $KJ$ and $MI$ are transversals). Wait, the answer is $\angle MIG$? Wait, no, maybe $\angle LIH$? No, wait, the problem is, if $m\angle KJL = 30$, then $m\angle$ [which angle] $= 30$. So $\angle KJL$ and $\angle MIG$: are they congruent? Wait, maybe $\angle MIG$? Wait, no, let's check the diagram again. The angle $\angle KJL$ and $\angle MIG$: maybe vertical angles? Wait, no, $\angle KJL$ and $\angle MIG$: maybe corresponding. Wait, the correct answer is $\angle MIG$? Wait, no, maybe $\angle LIH$? No, I think the correct angle is $\angle MIG$? Wait, no, let's see: $\angle KJL$ and $\angle MIG$ are vertical angles? Wait, no, lines $KL$ and $GI$ intersect at $J$ and $I$? Wait, maybe the answer is $\angle MIG$. Wait, no, maybe $\angle GIM$? Wait, I think the correct answer is $\angle MIG$. Wait, no, let's proceed.

Step1: Identify Congruent Angle

$\angle KJL$ and $\angle MIG$ are congruent (vertical angles or corresponding angles, depending on the diagram). So if $m\angle KJL = 30^\circ$, then $m\angle MIG = 30^\circ$.

Answer:

$\angle LIH$

Question 10