3)
6
x
4)
x=6
4
x
5)
40°
x
6)
x 76°
7)
54°
x
8)
x 75°
9)
68°
x
10)
20°
x

Question

Accepted Answer

### Problem 3:
# Explanation:
## Step1: Identify the triangle type (isosceles, as sides are marked equal)
The two triangles are similar or congruent? Wait, the segments are marked equal, so the two triangles are congruent? Wait, no, the line is an angle bisector? Wait, no, the two sides of the big triangle are marked equal, and the line divides them into equal parts. So by the Basic Proportionality Theorem (Thales' theorem), but wait, the two triangles are similar? Wait, no, the sides are marked equal, so the two triangles are congruent? Wait, the top triangle and the bottom triangle: the sides are marked equal, and the angle is common. So by SAS congruence, they are congruent. So $ x = 6 $.
# Answer:
$ x = 6 $

### Problem 4:
# Explanation:
## Step1: The triangle is isosceles (two sides marked equal), and the line is the perpendicular bisector? Wait, the vertical line is a median and angle bisector. So the two segments on the base are equal. So $ x = 4 $.
# Answer:
$ x = 4 $

### Problem 5:
# Explanation:
## Step1: The triangle is isosceles (two sides marked equal), so the base angles are equal. Let the base angles be $ x $.
## Step2: Sum of angles in a triangle is $ 180^\circ $. So $ 40^\circ + x + x = 180^\circ $
## Step3: Solve for $ x $: $ 2x = 180 - 40 = 140 $, so $ x = 70^\circ $
# Answer:
$ x = 70^\circ $

### Problem 6:
# Explanation:
## Step1: The triangle is isosceles (two sides marked equal), so the base angles are equal. Wait, the given angle is $ 76^\circ $, and $ x $ is the other base angle? Wait, no, the triangle has two equal sides, so the angles opposite them are equal. Wait, the angle at the base is $ 76^\circ $, so $ x = 76^\circ $? Wait, no, wait: sum of angles is $ 180^\circ $. Wait, maybe the triangle is isosceles with two equal sides, so the two base angles are equal. Wait, the angle given is $ 76^\circ $, so $ x = 76^\circ $? Wait, no, let's check: if two sides are equal, the angles opposite are equal. So if the triangle has sides marked equal, then the angles opposite are equal. So if one angle is $ 76^\circ $, and $ x $ is the other angle, then if $ 76^\circ $ is a base angle, then the other base angle is $ x $, and the vertex angle is $ 180 - 2*76 = 28^\circ $. But the diagram shows $ x $ and $ 76^\circ $ as base angles? Wait, maybe the triangle is isosceles with $ x $ and $ 76^\circ $ as base angles. So $ x = 76^\circ $? Wait, no, wait, maybe I got it wrong. Wait, the triangle has two equal sides, so the angles opposite are equal. So if the two equal sides are the legs, then the base angles are equal. So if one base angle is $ 76^\circ $, the other is $ x $, so $ x = 76^\circ $. Wait, but let's calculate: sum of angles is $ 180 $. So $ x + 76 + $ vertex angle $ = 180 $. But if it's isosceles with two equal sides, then two angles are equal. So if $ x $ and $ 76 $ are the equal angles, then $ x = 76 $. Wait, maybe the diagram shows that the two equal sides are the ones with the marks, so the angles opposite are equal. So $ x = 76^\circ $.
# Answer:
$ x = 76^\circ $

### Problem 7:
# Explanation:
## Step1: The triangle is isosceles (two sides marked equal), and the line is a midline? Wait, the line is parallel to the base? Wait, the angles: the exterior angle is $ 54^\circ $, so the interior angle is $ 180 - 54 = 126^\circ $? No, wait, the line is parallel to the base, and the triangle is isosceles. Wait, the line divides the sides into equal parts, so it's the midline. So the angle $ x $ is equal to the angle at the top? Wait, no, the triangle is isosceles, so the base angles are equal. Wait, the exterior angle is $ 54^\circ $, which is equal to the sum of the two remote interior angles. But the triangle is isosceles, so the two base angles are equal. Wait, maybe the line is parallel, so the angle $ x $ is equal to the angle at the top. Wait, the triangle has two equal sides, so the base angles are equal. Let's see: the line is parallel to the base, so the triangle formed by the line is similar to the original triangle. Since the sides are divided into equal parts, the ratio is $ 1:2 $, but the angles are equal. Wait, the exterior angle is $ 54^\circ $, which is equal to the sum of the two base angles? No, exterior angle is equal to the sum of the two non-adjacent interior angles. Wait, maybe the triangle is isosceles, so the base angles are equal, and the exterior angle is $ 54^\circ $, so the base angle is $ 54^\circ $, so $ x = 54^\circ $? Wait, no, maybe I'm overcomplicating. The line is a midline, so the triangle is similar, and the angle $ x $ is equal to the angle at the top. Wait, the triangle is isosceles, so the base angles are equal. The exterior angle is $ 54^\circ $, so the interior angle is $ 180 - 54 = 126^\circ $ (no, that's not right). Wait, maybe the angle $ 54^\circ $ is the base angle, so $ x = 54^\circ $.
# Answer:
$ x = 54^\circ $

### Problem 8:
# Explanation:
## Step1: The triangle is isosceles (two sides marked equal), so the base angles are equal. The exterior angle is $ 75^\circ $, so the interior angle is $ 180 - 75 = 105^\circ $? No, wait, the exterior angle is equal to the sum of the two non-adjacent interior angles. Since the triangle is isosceles, the two base angles are equal. Let the base angles be $ x $. So the exterior angle $ 75^\circ = x + x $ (since the two base angles are equal). Wait, no, exterior angle is equal to the sum of the two remote interior angles. So if the triangle is isosceles with two equal sides, then the two base angles are equal, and the exterior angle is equal to the sum of the two base angles. So $ 75^\circ = x + x $, so $ 2x = 75^\circ $, so $ x = 37.5^\circ $? Wait, no, that can't be. Wait, maybe the exterior angle is adjacent to the vertex angle. Wait, the triangle has two equal sides, so the base angles are equal. Let the vertex angle be $ y $, then the base angles are $ x $. So $ y + 2x = 180 $. The exterior angle is $ 180 - y = 75 $, so $ y = 180 - 75 = 105 $. Then $ 105 + 2x = 180 $, so $ 2x = 75 $, so $ x = 37.5^\circ $.
# Answer:
$ x = 37.5^\circ $

### Problem 9:
# Explanation:
## Step1: The two triangles are congruent (sides marked equal, and vertical angles are equal). The top triangle has an angle of $ 68^\circ $, so the bottom triangle also has an angle of $ 68^\circ $. The bottom triangle is isosceles (two sides marked equal), so the base angles are equal. Let $ x $ be the base angle.
## Step2: Sum of angles in a triangle: $ 68^\circ + x + x = 180^\circ $
## Step3: Solve: $ 2x = 180 - 68 = 112 $, so $ x = 56^\circ $
# Answer:
$ x = 56^\circ $

### Problem 10:
# Explanation:
## Step1: The triangle is isosceles (two sides marked equal), and there's a right angle and a $ 28^\circ $ angle. Wait, the diagram shows a right angle, so the other angle is $ 90 - 28 = 62^\circ $? No, wait, the triangle with $ x $ is isosceles, so $ x = 90 - 28 = 62^\circ $? Wait, the right angle is $ 90^\circ $, and the angle is $ 28^\circ $, so the angle $ x $ is $ 90 - 28 = 62^\circ $.
# Answer:
$ x = 62^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 3:

Step1: Identify the triangle type (isosceles, as sides are marked equal)

Step1: The triangle is isosceles (two sides marked equal), and the line is the perpendicular bisector? Wait, the vertical line is a median and angle bisector. So the two segments on the base are equal. So \( x = 4 \).

Step1: The triangle is isosceles (two sides marked equal), so the base angles are equal. Let the base angles be \( x \).

Step2: Sum of angles in a triangle is \( 180^\circ \). So \( 40^\circ + x + x = 180^\circ \)

Step3: Solve for \( x \): \( 2x = 180 - 40 = 140 \), so \( x = 70^\circ \)

Answer:

Problem 4: