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exam #2 geometry a
isosceles, equilateral, midsegment, and the pythagorean theorem
1.) find the value of x.
2.) find the value of x.
3.) find the value of x. ( m angle 2 = 12 x + 4 )
4.) find the missing length indicated.
find cd.

Question

Accepted Answer

### Problem 1: Find the value of $ x $
# Explanation:
## Step 1: Identify triangle type
The triangle is isosceles (two sides marked equal), so the base angles are equal? Wait, no—wait, the side with 82 and the angle $ x $: Wait, no, in an isosceles triangle, the angles opposite equal sides are equal. Wait, the two sides are marked equal, so the angles opposite them are equal. Wait, the angle given is 82, and the other angle at the base? Wait, no, the triangle has two equal sides, so it's isosceles with the two equal sides, so the base angles? Wait, no, maybe it's an isosceles triangle with one angle 82, and the other angle $ x $, and the vertex angle? Wait, no, sum of angles in a triangle is $ 180^\circ $. Wait, if two sides are equal, then the angles opposite them are equal. Wait, maybe the triangle has two equal sides, so the angles opposite are equal. Wait, the angle labeled $ x $ and the other angle? Wait, no, the side with 82 is a side, and the two equal sides are the other two. Wait, maybe it's an isosceles triangle with one angle 82, and the other two angles: but if two sides are equal, then two angles are equal. Wait, maybe the triangle is isosceles with the two equal sides, so the angles opposite are equal. Wait, perhaps the angle of 82 is one of the equal angles? Wait, no, let's think again. Wait, in a triangle, the sum of angles is $ 180^\circ $. If two sides are equal, then the angles opposite them are equal. So if the two equal sides are the legs, then the base angles are equal. Wait, maybe the triangle has angles: 82, 82, and $ x $? No, that can't be, because 82 + 82 + x = 180 → x = 16. Wait, that makes sense. So:

## Step 2: Sum of angles in triangle
Sum of angles in a triangle is $ 180^\circ $. Let the two equal angles be 82? Wait, no, maybe the angle $ x $ is the vertex angle, and the two base angles are equal. Wait, no, the side with 82 is a side, maybe the angle adjacent to 82 is equal to another angle. Wait, maybe the triangle is isosceles with two angles equal, and one angle is 82, and the other angle is $ x $, and the third angle is equal to 82? No, that would make sum 82 + 82 + x = 180 → x = 16. Yes, that's likely. So:

$ 82 + 82 + x = 180 $

## Step 3: Solve for $ x $
$ 164 + x = 180 $

$ x = 180 - 164 = 16 $

# Answer: $ \boldsymbol{16} $

### Problem 2: Find the value of $ x $
# Explanation:
## Step 1: Identify triangle type
The triangle is isosceles (two sides marked equal), so the angles opposite equal sides are equal. The given angle is 65, so the other base angle is also 65.

## Step 2: Sum of angles in triangle
Sum of angles in a triangle is $ 180^\circ $. So:

$ 65 + 65 + x = 180 $

## Step 3: Solve for $ x $
$ 130 + x = 180 $

$ x = 180 - 130 = 50 $

# Answer: $ \boldsymbol{50} $

### Problem 3: Find the value of $ x $ ( $ m\angle 2 = 12x + 4 $ )
# Explanation:
## Step 1: Identify triangle properties
The triangles are isosceles (marked with equal sides), and they are congruent? Wait, the two triangles are isosceles, and the vertical angles are equal. Wait, the lower triangle has an angle of 64, and the upper triangle has angle 2 ( $ m\angle 2 = 12x + 4 $ ). Since the triangles are isosceles and the sides are marked equal, the base angles of the lower triangle: since it's isosceles with two equal sides, the base angles are equal. Wait, the lower triangle has an angle of 64, and the other base angle is also 64? Wait, no, sum of angles in a triangle is $ 180^\circ $. So in the lower triangle, angles are 64, 64, and the vertex angle: 180 - 64 - 64 = 52. Then the vertical angle (opposite) is also 52. Then in the upper triangle, which is isosceles, the vertex angle is 52, so the base angles (angle 2 and the other angle) are equal. Wait, sum of angles in upper triangle: 52 + 2*(angle 2) = 180 → 2*(angle 2) = 128 → angle 2 = 64. Wait, but $ m\angle 2 = 12x + 4 $, so:

## Step 2: Set up equation
$ 12x + 4 = 64 $

## Step 3: Solve for $ x $
$ 12x = 64 - 4 = 60 $

$ x = \frac{60}{12} = 5 $

# Answer: $ \boldsymbol{5} $

### Problem 4: Find the missing length $ CD $
# Explanation:
## Step 1: Identify midsegment theorem
The midsegment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. In the diagram, $ C $ and $ D $ are midpoints (since the sides are marked with equal segments, indicating midpoints). So $ CD $ is the midsegment of triangle $ ABE $, and $ BE = 30 $.

## Step 2: Apply midsegment theorem
Midsegment length is half the length of the third side. So $ CD = \frac{1}{2} \times BE $

## Step 3: Calculate $ CD $
$ CD = \frac{1}{2} \times 30 = 15 $

# Answer: $ \boldsymbol{15} $

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QUESTION IMAGE

Question

Explanation:

Problem 1: Find the value of \( x \)

Step 1: Identify triangle type

Step 2: Sum of angles in triangle

Step 3: Solve for \( x \)

Step 1: Identify triangle type

Step 2: Sum of angles in triangle

Step 3: Solve for \( x \)

Step 1: Identify triangle properties

Step 2: Set up equation

Step 3: Solve for \( x \)

Answer:

Problem 2: Find the value of \( x \)