two parallel lines are cut by a transversal as shown below. suppose m∠8 = 136°. find m∠1 and m∠3.

Question

Accepted Answer

# Explanation:
## Step1: Find $ m\angle1 $
$ \angle1 $ and $ \angle8 $ are corresponding angles (since two parallel lines cut by a transversal, corresponding angles are equal). Wait, no, actually $ \angle8 $ and $ \angle5 $ are adjacent, and $ \angle1 $ and $ \angle5 $? Wait, maybe better to use linear pair or alternate interior. Wait, $ \angle8 $ and $ \angle5 $ are adjacent? No, $ \angle8 $ and $ \angle6 $ are vertical? Wait, no, let's look at the diagram. The two parallel lines, transversal. $ \angle8 $ and $ \angle1 $: Wait, $ \angle8 $ and $ \angle4 $? No, maybe $ \angle8 $ and $ \angle2 $? Wait, no, let's recall: when two parallel lines are cut by a transversal, consecutive interior angles are supplementary, corresponding angles are equal, alternate interior are equal, vertical angles are equal.

Wait, $ \angle8 $ and $ \angle5 $ are adjacent? No, $ \angle8 $ and $ \angle6 $ are vertical angles? Wait, no, $ \angle8 $ and $ \angle5 $ are a linear pair? Wait, $ \angle8 + \angle5 = 180^\circ $? No, $ \angle8 $ and $ \angle7 $ are vertical? Wait, maybe I made a mistake. Let's see: $ \angle8 $ and $ \angle1 $: Wait, $ \angle8 $ and $ \angle1 $ are corresponding angles? Wait, no, the two parallel lines: let's label the lines as line $ l $ (left) and line $ k $ (right), transversal $ t $. Then $ \angle8 $ is on line $ l $, below the transversal, and $ \angle1 $ is on line $ k $, above the transversal. Wait, maybe $ \angle8 $ and $ \angle2 $ are corresponding? No, maybe $ \angle8 $ and $ \angle4 $ are alternate interior? Wait, no, let's think about vertical angles and linear pairs.

Wait, $ \angle8 $ and $ \angle6 $ are vertical angles? No, $ \angle8 $ and $ \angle5 $ are adjacent, forming a linear pair, so $ \angle8 + \angle5 = 180^\circ $? Wait, no, $ \angle8 $ and $ \angle7 $ are vertical angles, so $ m\angle7 = m\angle8 = 136^\circ $? No, vertical angles are equal, but $ \angle8 $ and $ \angle6 $ are vertical? Wait, maybe the diagram: the transversal intersects the two parallel lines, creating 8 angles. Let's number them: top line (right) has angles 1,2,3,4 (1 top, 2 bottom, 3 bottom, 4 top? Wait, no, standard numbering: when transversal cuts two parallel lines, the angles are numbered 1,2,3,4 on the top line (from top to bottom, left to right), and 5,6,7,8 on the bottom line (left to right, top to bottom). So $ \angle1 $ and $ \angle5 $ are corresponding angles, $ \angle2 $ and $ \angle6 $, $ \angle3 $ and $ \angle7 $, $ \angle4 $ and $ \angle8 $. Ah! There we go. So $ \angle4 $ and $ \angle8 $ are corresponding angles, so $ m\angle4 = m\angle8 = 136^\circ $. Then $ \angle1 $ and $ \angle4 $ are a linear pair (they form a straight line), so $ m\angle1 + m\angle4 = 180^\circ $. Therefore, $ m\angle1 = 180^\circ - 136^\circ = 44^\circ $.

Then $ \angle3 $ and $ \angle1 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $: $ \angle1 $ and $ \angle3 $ are vertical angles? No, $ \angle1 $ and $ \angle3 $: $ \angle1 $ and $ \angle2 $ are a linear pair, $ \angle2 $ and $ \angle3 $ are vertical? Wait, no, $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $: if $ \angle1 $ is at the top right, $ \angle3 $ is at the bottom right, so they are vertical angles? No, $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $: $ \angle1 $ and $ \angle2 $ are adjacent, $ \angle2 $ and $ \angle3 $ are adjacent, so $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, no, vertical angles are opposite each other when two lines intersect. So the transversal intersects the right parallel line, forming $ \angle1, \angle2, \angle3, \angle4 $. So $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $: $ \angle1 $ is top, $ \angle3 $ is bottom, so yes, vertical angles? Wait, no, $ \angle1 $ and $ \angle3 $: if the transversal intersects the right line, then $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, no, $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $: $ \angle1 $ and $ \angle2 $ are a linear pair (sum to 180), $ \angle2 $ and $ \angle3 $ are a linear pair? No, that can't be. Wait, no, when two lines intersect, vertical angles are equal. So the transversal and the right parallel line intersect, forming four angles: $ \angle1 $ (top), $ \angle2 $ (bottom right), $ \angle3 $ (bottom left), $ \angle4 $ (top left). So $ \angle1 $ and $ \angle3 $ are vertical angles? No, $ \angle1 $ and $ \angle3 $ are opposite? Wait, $ \angle1 $ and $ \angle3 $: $ \angle1 $ is top, $ \angle3 $ is bottom left, $ \angle2 $ is bottom right, $ \angle4 $ is top left. So $ \angle1 $ and $ \angle3 $ are vertical angles? No, $ \angle1 $ and $ \angle3 $ are not vertical. Wait, $ \angle1 $ and $ \angle3 $ are alternate interior? No, the two parallel lines: right line and left line. $ \angle3 $ is on the right line, bottom left, and $ \angle5 $ is on the left line, top left. So $ \angle3 $ and $ \angle5 $ are alternate interior angles, so they are equal. And $ \angle5 $ and $ \angle8 $ are a linear pair? Wait, $ \angle5 + \angle8 = 180^\circ $, so $ \angle5 = 180^\circ - 136^\circ = 44^\circ $. Then $ \angle3 = \angle5 = 44^\circ $ (alternate interior angles). And $ \angle1 $ and $ \angle3 $: wait, $ \angle1 $ and $ \angle3 $ are vertical angles? No, $ \angle1 $ and $ \angle3 $: wait, $ \angle1 $ and $ \angle2 $ are a linear pair, $ \angle2 $ and $ \angle3 $ are a linear pair? No, that's not possible. Wait, maybe $ \angle1 $ and $ \angle3 $ are equal? Wait, no, let's go back.

Wait, $ m\angle8 = 136^\circ $. $ \angle8 $ and $ \angle5 $ are a linear pair (they form a straight line), so $ m\angle5 = 180^\circ - 136^\circ = 44^\circ $. $ \angle5 $ and $ \angle1 $ are corresponding angles (since two parallel lines cut by a transversal), so $ m\angle1 = m\angle5 = 44^\circ $. Then $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, no, $ \angle1 $ and $ \angle3 $: $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, the transversal intersects the right parallel line, so $ \angle1 $ (top) and $ \angle3 $ (bottom left) are vertical? No, $ \angle1 $ and $ \angle3 $ are not vertical. Wait, $ \angle1 $ and $ \angle3 $ are alternate interior? No, $ \angle3 $ and $ \angle5 $ are alternate interior, so $ \angle3 = \angle5 = 44^\circ $, and $ \angle1 = \angle5 = 44^\circ $ (corresponding). So $ m\angle1 = 44^\circ $, $ m\angle3 = 44^\circ $? Wait, but let's check with linear pair. $ \angle1 $ and $ \angle4 $ are a linear pair, so $ m\angle1 + m\angle4 = 180^\circ $. $ m\angle4 = m\angle8 = 136^\circ $ (corresponding angles), so $ m\angle1 = 180 - 136 = 44^\circ $. Then $ \angle3 $ and $ \angle1 $: $ \angle3 $ and $ \angle1 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $ are vertical angles? No, $ \angle1 $ and $ \angle3 $ are not vertical. Wait, $ \angle3 $ and $ \angle1 $ are equal because they are corresponding? No, $ \angle3 $ and $ \angle5 $ are alternate interior, $ \angle5 = 44^\circ $, so $ \angle3 = 44^\circ $, and $ \angle1 = 44^\circ $ (corresponding to $ \angle5 $). So both $ m\angle1 = 44^\circ $ and $ m\angle3 = 44^\circ $.

## Step1: Find $ m\angle1 $
$ \angle1 $ and $ \angle8 $ are not corresponding, but $ \angle8 $ and $ \angle4 $ are corresponding (since two parallel lines cut by transversal, corresponding angles are equal). So $ m\angle4 = m\angle8 = 136^\circ $. $ \angle1 $ and $ \angle4 $ form a linear pair (they are adjacent and form a straight line), so $ m\angle1 + m\angle4 = 180^\circ $. Thus, $ m\angle1 = 180^\circ - 136^\circ = 44^\circ $.

## Step2: Find $ m\angle3 $
$ \angle3 $ and $ \angle1 $ are vertical angles (when two lines intersect, vertical angles are equal). Wait, no, $ \angle3 $ and $ \angle1 $: wait, the transversal intersects the right parallel line, so $ \angle1 $ (top) and $ \angle3 $ (bottom left) are vertical angles? Wait, no, $ \angle1 $ and $ \angle3 $ are vertical angles? Wait, $ \angle1 $ and $ \angle3 $ are opposite angles when the transversal intersects the right parallel line, so yes, vertical angles. Therefore, $ m\angle3 = m\angle1 = 44^\circ $. Alternatively, $ \angle3 $ and $ \angle5 $ are alternate interior angles (since two parallel lines cut by transversal), and $ \angle5 $ and $ \angle8 $ are a linear pair, so $ m\angle5 = 180^\circ - 136^\circ = 44^\circ $, thus $ m\angle3 = m\angle5 = 44^\circ $.

# Answer:
$ m\angle1 = 44^\circ $, $ m\angle3 = 44^\circ $

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Explanation:

Step1: Find \( m\angle1 \)

Answer: