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5. in jada’s sketch, ( mangle 1 = mangle 4 ). what is ( mangle 5 )?
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# Explanation:
## Step1: Find $ m\angle2 $
In a triangle, the sum of angles is $ 180^\circ $. For the triangle with $ 85^\circ $, $ 80^\circ $, and $ \angle2 $:
$ m\angle2 = 180^\circ - 85^\circ - 80^\circ = 15^\circ $

## Step2: Find $ m\angle3 $
Given $ m\angle3 = 90^\circ $ (from the diagram).

## Step3: Find $ m\angle4 $
In a triangle (or straight - line related angles), the sum of angles around a point or in a triangle. First, consider the angles on a straight line or in the triangle with $ \angle2 $, $ \angle3 $, and the angle adjacent to $ \angle4 $. Wait, actually, since $ m\angle1=m\angle4 $, and we can find $ m\angle1 $ first? Wait, no, let's use the fact that in the triangle with $ 50^\circ $, and the angle related to $ \angle5 $, but maybe better to use the sum of angles in the quadrilateral or the triangle involving $ \angle2 $, $ \angle3 $, and the angle equal to $ \angle1 $. Wait, actually, let's find $ m\angle1 $ first. Wait, no, let's look at the angles around the intersection. Wait, the angles $ \angle1 $, $ 80^\circ $, and $ 85^\circ $: no, wait, the triangle with $ 85^\circ $, $ 80^\circ $, and $ \angle2 $ we did. Now, the angles on a straight line: the sum of angles on a straight line is $ 180^\circ $. For the line with $ \angle1 $, $ 80^\circ $, and the other angle? Wait, maybe another approach. Let's find the angle that is equal to $ \angle1 $. Wait, $ m\angle1 $: in the triangle with $ 85^\circ $ and $ 80^\circ $, we found $ m\angle2 = 15^\circ $. Then, in the triangle with $ \angle2 $, $ \angle3 = 90^\circ $, and the angle $ x $ (let's say), $ x=180^\circ - 90^\circ - 15^\circ = 75^\circ $. Then, since $ m\angle1=m\angle4 $, and $ m\angle1 $ is equal to $ 75^\circ $ (because of vertical angles or corresponding angles? Wait, maybe I made a mistake. Wait, let's start over.

First, in the triangle with angles $ 85^\circ $, $ 80^\circ $, and $ \angle2 $:
$ m\angle2=180 - 85 - 80=15^\circ $

Then, in the triangle with $ \angle2 = 15^\circ $, $ \angle3 = 90^\circ $, the third angle (let's call it $ \angle a $) is $ 180 - 90 - 15 = 75^\circ $

Since $ m\angle1=m\angle4 $, and $ \angle1 $ is equal to $ \angle a = 75^\circ $ (because of alternate interior angles or vertical angles, assuming parallel lines). So $ m\angle4 = 75^\circ $

Now, look at the triangle with $ 50^\circ $, $ \angle5 $, and the angle equal to $ \angle4 $ (or $ \angle1 $). Wait, the sum of angles in a triangle is $ 180^\circ $. The triangle has angles $ 50^\circ $, $ \angle5 $, and $ \angle4 $ (since $ m\angle1 = m\angle4 $ and they are corresponding or alternate angles). Wait, no, the triangle with $ \angle5 $, $ 50^\circ $, and the angle equal to $ \angle4 $. So:

$ m\angle5=180^\circ - 50^\circ - m\angle4 $

Since $ m\angle4 = 75^\circ $ (from above), then:

$ m\angle5=180 - 50 - 75 = 55^\circ $? Wait, no, that can't be. Wait, maybe the triangle is different. Wait, let's look at the diagram again. The angle $ 50^\circ $, $ \angle5 $, and the angle that is equal to $ \angle1 $ (since $ m\angle1 = m\angle4 $). Wait, maybe the sum of angles in the triangle with $ \angle5 $, $ 50^\circ $, and $ \angle1 $ (since $ m\angle1 = m\angle4 $) is $ 180^\circ $. Wait, let's find $ m\angle1 $ correctly.

In the triangle with angles $ 85^\circ $, $ 80^\circ $, and $ \angle2 $: $ m\angle2 = 15^\circ $

Then, the angle adjacent to $ \angle3 = 90^\circ $ and $ \angle2 = 15^\circ $ is $ 180 - 90 - 15 = 75^\circ $. This angle is equal to $ m\angle1 $ (because they are corresponding angles, assuming the lines are parallel). So $ m\angle1 = 75^\circ $, so $ m\angle4 = 75^\circ $

Now, look at the triangle that has $ \angle5 $, $ 50^\circ $, and $ \angle4 $. Wait, no, the triangle with $ \angle5 $, $ 50^\circ $, and the angle equal to $ \angle1 $ (which is $ 75^\circ $). So:

$ m\angle5=180 - 50 - 75 = 55^\circ $? Wait, no, maybe I messed up the triangle. Wait, another approach: the sum of angles around a point is $ 360^\circ $, but maybe not. Wait, let's use the fact that in the diagram, the angle $ \angle5 $, $ 50^\circ $, and the angle equal to $ \angle1 $ (which is $ m\angle4 $) form a triangle. Wait, maybe the correct way is:

First, find $ m\angle1 $: in the triangle with $ 85^\circ $ and $ 80^\circ $, $ m\angle1 = 180 - 85 - 80 = 15^\circ $? No, that's wrong. Wait, no, the triangle with $ 85^\circ $, $ 80^\circ $, and $ \angle2 $: $ \angle2 = 15^\circ $. Then, the angle next to $ \angle3 = 90^\circ $ and $ \angle2 = 15^\circ $ is $ 75^\circ $, which is equal to $ m\angle1 $ (because they are alternate interior angles). So $ m\angle1 = 75^\circ $, so $ m\angle4 = 75^\circ $

Now, the triangle with $ \angle5 $, $ 50^\circ $, and $ \angle4 $: sum of angles is $ 180^\circ $, so $ m\angle5 = 180 - 50 - 75 = 55^\circ $? Wait, no, maybe the triangle is $ \angle5 $, $ 50^\circ $, and $ \angle1 $. Wait, $ m\angle1 = 75^\circ $, $ 50^\circ $, so $ m\angle5 = 180 - 50 - 75 = 55^\circ $. Wait, but let's check again.

Wait, maybe the correct steps are:

1. Find $ m\angle2 $: In the triangle with angles $ 85^\circ $, $ 80^\circ $, and $ \angle2 $, $ m\angle2=180 - 85 - 80 = 15^\circ $

2. Find $ m\angle4 $: In the triangle with $ \angle2 = 15^\circ $, $ \angle3 = 90^\circ $, the third angle is $ 180 - 90 - 15 = 75^\circ $. Since $ m\angle1 = m\angle4 $, and $ m\angle1 $ is equal to this $ 75^\circ $ angle (corresponding angles), so $ m\angle4 = 75^\circ $

3. Find $ m\angle5 $: In the triangle with angles $ 50^\circ $, $ m\angle4 = 75^\circ $, and $ m\angle5 $, we have $ m\angle5=180 - 50 - 75 = 55^\circ $

Wait, but maybe I made a mistake in the triangle. Alternatively, maybe the triangle has angles $ 50^\circ $, $ m\angle5 $, and $ m\angle1 $, and since $ m\angle1 = m\angle4 $, and we found $ m\angle1 = 75^\circ $, then $ m\angle5 = 180 - 50 - 75 = 55^\circ $

# Answer:
$ 55^\circ $

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QUESTION IMAGE

Question

Explanation:

Step1: Find \( m\angle2 \)

Step2: Find \( m\angle3 \)

Step3: Find \( m\angle4 \)

Answer: