Triangle XYZ is an isosceles triangle with sides XY = XZ. If the measure of angle Y is 80°, what is the measure of angle X?

Correct — D. The equal sides are XY = XZ, so X is the apex and YZ is the base.

Angles opposite equal sides are equal. Side XY faces vertex Z and side XZ faces vertex Y, so:
∠Z = ∠Y = 80°

Angle sum of the triangle:
∠X = 180° − 80° − 80° = 20° → option (d).

Why the others are wrong

• (a)
  30° — 30° needs base angles of 75° each. The stem prints ∠Y = 80°, and ∠Z has to match it exactly.
• (b)
  40° — 40° is just half of 80°. No isosceles rule makes the apex half a base angle; the 180° sum is the only relation available here.
• (c)
  50° — 50° answers the mirrored question: if 80° were the apex angle X, each base angle would be (180 − 80) ⁄ 2 = 50°. Here 80° sits at Y, a base angle.

The base-angle theorem says angles opposite equal sides are equal. Its converse is equally usable: equal angles force equal sides.

The only real skill is reading which angles those are. In △XYZ, side XY is opposite vertex Z and side XZ is opposite vertex Y, so XY = XZ pairs ∠Z with ∠Y — not ∠X with ∠Y.

Once the pair is right, the 180° sum finishes the question in one subtraction.

Key facts

• Angles opposite equal sides of a triangle are equal, and the converse also holds.
• In △XYZ, side XY faces vertex Z and side XZ faces vertex Y, so XY = XZ gives ∠Z = ∠Y.
• The three angles of a triangle sum to 180°, so two 80° base angles leave 20° at the apex.
• Each base angle of an isosceles triangle must stay below 90°, since two of them alone cannot reach 180°.

Common traps

• Assuming the given 80° is the apex angle X
• Pairing the equal sides with the wrong vertices, reading XY = XZ as ∠X = ∠Y
• Halving or doubling the given angle instead of using the 180° sum

No directly related past PYQ was found.