Question

1.) find the value of x.
2.) find the value of x.
3.) find the value of x. m∠2 = 12x + 4
4.) find the missing length indicated.
find cd.

Explanation:

Problem 1:

Step1: Identify triangle type (isosceles)

The triangle has two equal sides (marked with ticks), so it's isosceles. Thus, the two base angles are equal? Wait, no—wait, the side with 82 and the side opposite? Wait, no, the triangle has two equal sides (the two sides with ticks), so the angles opposite them are equal? Wait, no, in the first triangle, the two sides with ticks are the legs? Wait, no, the angle labeled 82—wait, maybe it's an isosceles triangle with two equal sides, so the base angles? Wait, no, the angle at the end is x, and the side with 82—wait, maybe it's an isosceles triangle where the two equal sides are the ones with ticks, so the base angles? Wait, no, maybe the triangle is isosceles with the two sides (the ones with ticks) being equal, so the angles opposite them are equal. Wait, but one angle is 82, and the other is x. Wait, no—wait, maybe it's a triangle with two equal sides, so it's isosceles, so the base angles are equal? Wait, no, maybe the angle of 82 is one of the equal angles? Wait, no, the triangle has two equal sides (marked with ticks), so the angles opposite those sides are equal. Wait, the side with 82—maybe that's a leg, and the other leg is equal (ticks), so the base angles? Wait, no, maybe the triangle is isosceles with the two sides (ticks) being equal, so the angles opposite are equal. Wait, the angle at the vertex with x is the vertex angle, and the other two angles are equal? Wait, no, the angle labeled 82—maybe that's one of the base angles, and x is the other base angle? But that would mean 82 = x, but that doesn't make sense. Wait, no—wait, maybe it's a triangle where the two equal sides are the ones with ticks, so the base angles are equal, but the angle of 82 is the vertex angle? Wait, no, the sum of angles in a triangle is 180. So if it's isosceles with two equal angles, then 82 + 82 + x = 180? Wait, no, maybe the angle of 82 is one of the equal angles, and x is the vertex angle. Wait, let's re-examine: the triangle has two sides marked with ticks (equal), so it's isosceles. The angle adjacent to the side labeled 82—maybe that's a right angle? No, the triangle is not right-angled. Wait, maybe the angle of 82 is one of the base angles, and x is the vertex angle. So sum of angles: 82 + 82 + x = 180. Then x = 180 - 82 - 82 = 16. Wait, that makes sense. So step 1: identify it's an isosceles triangle (two equal sides, ticks). Step 2: sum of angles in triangle is 180. Step 3: two equal angles (82 each), so x = 180 - 2*82 = 180 - 164 = 16.

Step1: Identify triangle type (isosceles)

The triangle has two equal sides (marked with ticks), so it is isosceles with two equal angles.

Step2: Sum of angles in a triangle

The sum of the interior angles of a triangle is \( 180^\circ \).

Step3: Calculate x

Let the two equal angles be \( 82^\circ \) each. Then:
\( 82 + 82 + x = 180 \)
\( 164 + x = 180 \)
\( x = 180 - 164 \)
\( x = 16 \)

Step1: Identify triangle type (isosceles)

The triangle has two sides marked with ticks (equal), so it is isosceles. Thus, the two base angles are equal.

Step2: Sum of angles in a triangle

Sum of interior angles in a triangle is \( 180^\circ \). One base angle is \( 65^\circ \), so the other base angle is also \( 65^\circ \).

Step3: Calculate x (vertex angle)

\( x + 65 + 65 = 180 \)
\( x + 130 = 180 \)
\( x = 180 - 130 \)
\( x = 50 \)

Step1: Identify congruent triangles (vertical angles, equal sides)

The two triangles are congruent? Wait, the triangles have vertical angles (equal), and sides marked with ticks (equal). So by SAS or ASA? Wait, the top triangle has sides with ticks (equal) and the bottom triangle has sides with ticks (equal). The vertical angles are equal. So the triangles are congruent. Thus, angle 2 is equal to 64? Wait, no—wait, the bottom triangle has an angle of 64, and the top triangle has angle 2. Since the triangles are congruent (vertical angles, equal sides), angle 2 = 64? But \( m\angle 2 = 12x + 4 \). So set \( 12x + 4 = 64 \).

Step2: Solve for x

\( 12x + 4 = 64 \)
Subtract 4: \( 12x = 60 \)
Divide by 12: \( x = 5 \)

Answer:

\( x = 16 \)

Problem 2: