find the measure of the missing angle or solve for x.
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find fg

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find hj

Question

find the measure of the missing angle or solve for x.
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2.

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8.

9.
find fg

10.
find hj

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Recall triangle angle sum (180°)
The sum of angles in a triangle is $ 180^\circ $. Let the missing angle be $ y $. So, $ 65^\circ + 51^\circ + y = 180^\circ $.
## Step2: Calculate the missing angle
First, add $ 65^\circ $ and $ 51^\circ $: $ 65 + 51 = 116 $. Then, $ y = 180 - 116 = 64^\circ $.
# Answer: $ 64^\circ $

### Problem 2:
# Explanation:
## Step1: Identify right triangle angles
This is a right triangle (one angle $ 90^\circ $), and another angle is $ 60^\circ $. Let the missing angle be $ z $.
## Step2: Use angle sum (180°)
$ 90^\circ + 60^\circ + z = 180^\circ $. So, $ z = 180 - 90 - 60 = 30^\circ $.
# Answer: $ 30^\circ $

### Problem 3:
# Explanation:
## Step1: Recognize isosceles triangle
The triangle is isosceles (two equal sides), so the base angles are equal. The vertex angle is $ 48^\circ $, so the base angles sum to $ 180 - 48 = 132^\circ $. Each base angle is $ \frac{132}{2} = 66^\circ $.
## Step2: Find the exterior angle
The exterior angle is equal to the sum of the two non - adjacent interior angles. But here, the exterior angle is supplementary to the base angle? Wait, no. Wait, the triangle is isosceles with vertex angle $ 48^\circ $, so base angles are $ 66^\circ $. The exterior angle (adjacent to the base angle) is $ 180 - 66 = 114^\circ $? Wait, no, the diagram shows a triangle with a horizontal line extended from the base. Wait, maybe the triangle is isosceles, so the base angles are equal. Let the base angle be $ x $. Wait, the problem says "solve for $ x $". Wait, maybe the triangle is isosceles, so the two base angles are equal. The vertex angle is $ 48^\circ $, so $ 2x + 48 = 180 $, $ 2x = 132 $, $ x = 66^\circ $. But then the exterior angle (the angle outside the triangle at the base) would be $ 180 - 66 = 114^\circ $? Wait, maybe I misread. Wait, the triangle has a mark indicating two equal sides, so it's isosceles with vertex angle $ 48^\circ $. So the base angles are $ (180 - 48)/2 = 66^\circ $. If the exterior angle is adjacent to the base angle, then $ x = 180 - 66 = 114^\circ $? Wait, no, maybe the angle $ x $ is the exterior angle. Wait, the triangle has a horizontal line, so the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, the two non - adjacent interior angles are the vertex angle $ 48^\circ $ and the other base angle $ 66^\circ $? No, that's not right. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. In an isosceles triangle with vertex angle $ 48^\circ $, the two base angles are $ 66^\circ $ each. So the exterior angle (at the base) is $ 48^\circ+66^\circ = 114^\circ $. So $ x = 114^\circ $.
# Answer: $ 114^\circ $

### Problem 4:
# Explanation:
## Step1: Identify the angle relationship
There is a right angle (from the perpendicular line) and a $ 47^\circ $ angle. The angle $ x $ and $ 47^\circ $ are complementary (since they form a right angle? Wait, no, the diagram shows a triangle with a horizontal line and a perpendicular line, so the angle between the perpendicular and the other side is $ x $, and the angle with the horizontal is $ 47^\circ $. So $ x + 47^\circ=90^\circ $ (if it's a right triangle). Wait, no, maybe it's a right triangle, so $ x = 90 - 47 = 43^\circ $.
## Step2: Calculate $ x $
$ x = 90 - 47 = 43^\circ $.
# Answer: $ 43^\circ $

### Problem 5:
# Explanation:
## Step1: Recognize isosceles triangle and exterior angle
The triangle is isosceles (marked sides), so the base angles are equal. The exterior angle is $ 40^\circ $, and by the exterior angle theorem, the exterior angle is equal to the sum of the two non - adjacent interior angles. Since the triangle is isosceles, the two non - adjacent interior angles are equal (the base angles). Let each base angle be $ x $. So $ 2x = 40^\circ $? No, that's not right. Wait, the exterior angle is supplementary to the adjacent interior angle. Wait, the exterior angle is $ 40^\circ $, so the adjacent interior angle is $ 180 - 40 = 140^\circ $. But in an isosceles triangle, the vertex angle can't be $ 140^\circ $ if the base angles are equal? Wait, no, maybe the angle $ x $ is the base angle. Wait, the diagram shows a triangle with a horizontal line and an exterior angle of $ 40^\circ $. Wait, maybe the triangle is isosceles, so the two base angles are equal. The exterior angle is equal to the sum of the two remote interior angles. So if the exterior angle is $ 40^\circ $, and the two remote interior angles are equal (since it's isosceles), then each is $ 20^\circ $? No, that doesn't make sense. Wait, maybe I got the exterior angle wrong. Wait, the angle given is $ 40^\circ $, and the triangle is isosceles with two equal sides. So the base angles are equal. Let the base angle be $ x $, then the exterior angle (adjacent to the base angle) is $ 180 - x $. But if the exterior angle is $ 40^\circ $, then $ 180 - x = 40 $, so $ x = 140^\circ $, which is impossible for a triangle (since the sum of angles would exceed $ 180^\circ $). Wait, maybe the angle $ 40^\circ $ is the vertex angle's exterior? No, maybe the triangle is isosceles, and the angle $ x $ is the base angle, and the exterior angle is $ 40^\circ $, so by exterior angle theorem, $ x + x = 40^\circ $ (since the two base angles are equal and the exterior angle is equal to their sum). So $ 2x = 40 $, $ x = 20^\circ $. Yes, that makes sense. The exterior angle is equal to the sum of the two non - adjacent interior angles, and since the triangle is isosceles, the two non - adjacent interior angles (the base angles) are equal. So $ x + x = 40^\circ $, so $ x = 20^\circ $.
# Answer: $ 20^\circ $

### Problem 6:
# Explanation:
## Step1: Recognize equilateral triangle
The triangle has all three sides marked as equal, so it's an equilateral triangle. In an equilateral triangle, all angles are $ 60^\circ $. So $ x = 60^\circ $.
# Answer: $ 60^\circ $

### Problem 7:
# Explanation:
## Step1: Recognize midline theorem (triangle midsegment)
The segment $ RS $ is a midsegment of triangle $ ABC $ (since $ R $ and $ S $ are midpoints of $ AB $ and $ BC $ respectively). The midline theorem states that the midsegment is parallel to the third side and half its length. So $ RS=\frac{1}{2}AC $.
## Step2: Solve for $ x $
Given $ RS = 56 $, so $ 56=\frac{1}{2}x $, then $ x = 56\times2 = 112 $.
# Answer: $ 112 $

### Problem 8:
# Explanation:
## Step1: Recognize midline theorem
The segment $ HI $ is a midsegment of triangle $ CDE $ (since $ H $ and $ I $ are midpoints of $ CD $ and $ DE $ respectively). By the midline theorem, $ HI=\frac{1}{2}CE $.
## Step2: Calculate $ x $
Given $ CE = 142 $, so $ x=\frac{1}{2}\times142 = 71 $.
# Answer: $ 71 $

### Problem 9:
# Explanation:
## Step1: Recognize similar triangles (midline theorem)
The segment $ FG $ is a midsegment of triangle $ RST $ (since $ F $ and $ G $ are midpoints of $ RS $ and $ ST $ respectively). The midline theorem states that the midsegment is parallel to the third side and half its length. Wait, the length of $ RT = 112 $, $ RS = 39 + 39 = 78 $? No, wait, the marks indicate that $ F $ is the midpoint of $ RS $ and $ G $ is the midpoint of $ ST $. So $ FG=\frac{1}{2}RT $.
## Step2: Calculate $ FG $
Given $ RT = 112 $, so $ FG=\frac{1}{2}\times112 = 56 $. Wait, but let's check the sides. $ RS $ has length $ 39\times2 = 78 $, $ ST $ has length $ 86\div2\times3 $? No, maybe the triangles are similar. The ratio of $ RS $ to $ RT $ - no, better to use midline. Since $ F $ and $ G $ are midpoints, $ FG $ is midline, so $ FG=\frac{1}{2}RT $. $ RT = 112 $, so $ FG = 56 $.
# Answer: $ 56 $

### Problem 10:
# Explanation:
## Step1: Recognize midline or segment division
The segment $ HI $ is divided into three equal parts (marks), and the triangle has equal - length segments. The length $ HI = 110 $, and we need to find $ HJ $. Wait, the diagram shows $ H $, $ G $, $ I $ with three equal segments, and the triangle has equal sides. Wait, maybe $ HJ $ is related to the midline. Wait, the length $ HI = 110 $, and the triangle has a side with length $ 85 $? No, maybe the segment $ HJ $ is equal to $ 85 $? Wait, no, let's look at the marks. The segment $ HI $ is divided into three equal parts, and the other side has equal marks. Wait, maybe it's a midline or congruent segments. Wait, the problem says "Find $ HJ $". The diagram shows $ H $, $ G $, $ I $ with three equal segments (length $ 110 $), so each segment is $ 110\div3 $? No, maybe the triangle is isosceles, and $ HJ = 85 $. Wait, no, maybe the length $ HJ $ is equal to $ 85 $ because of congruent segments. Alternatively, since the marks indicate equal segments, $ HJ = 85 $. Wait, maybe the answer is $ 85 $.
# Answer: $ 85 $ (Note: This problem's solution is based on the visual marks indicating congruent segments, assuming $ HJ $ is congruent to the given segment of length $ 85 $.)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Recall triangle angle sum (180°)

Step2: Calculate the missing angle

Step1: Identify right triangle angles

Step2: Use angle sum (180°)

Step1: Recognize isosceles triangle

Step2: Find the exterior angle

Answer:

Problem 2: