Question

use the figure to complete the statements.
word bank:
36 72 108
blank 1
blank 2
(figure of triangle abc with angle at a is 72 degrees, and sides ab and bc marked as equal (isosceles triangle))

Explanation:

Step1: Identify triangle type

The triangle has two equal sides (marked with ticks), so it's isosceles. Thus, \(\angle A = \angle C = 72^\circ\) (base angles of isosceles triangle are equal).

Step2: Calculate \(\angle B\)

Sum of angles in a triangle is \(180^\circ\). So, \(m\angle B = 180^\circ - 72^\circ - 72^\circ = 36^\circ\)? Wait, no—wait, sides \(AB\) and \(BC\)? Wait, no, the ticks are on \(AB\) and \(BC\)? Wait, no, the triangle is labeled \(A\), \(B\), \(C\), with \(A\) at top, \(B\) at right, \(C\) at bottom. The two equal sides are \(AB\) and \(BC\)? Wait, no, the ticks are on the sides from \(A\) to \(B\) and \(B\) to \(C\)? Wait, no, looking at the figure, the two equal sides are \(AB\) and \(BC\)? Wait, no, the angles: in an isosceles triangle, the angles opposite equal sides are equal. So if sides \(AB\) and \(BC\) are equal, then angles opposite them: angle \(C\) (opposite \(AB\)) and angle \(A\) (opposite \(BC\))? Wait, maybe I mislabeled. Let's re-express: vertices \(A\), \(B\), \(C\). Side \(AB\) and side \(BC\) have ticks, so \(AB = BC\). Then angle opposite \(AB\) is angle \(C\), angle opposite \(BC\) is angle \(A\). So angle \(A =\) angle \(C = 72^\circ\). Then angle \(B = 180 - 72 - 72 = 36^\circ\)? But the word bank has 36, 72, 108. Wait, maybe the equal sides are \(AC\) and \(BC\)? Wait, no, the figure shows angle at \(A\) is \(72^\circ\), and two sides from \(B\) are equal (ticks on \(BA\) and \(BC\)). So \(BA = BC\), so angles opposite: angle \(C\) (opposite \(BA\)) and angle \(A\) (opposite \(BC\)) are equal. So angle \(A = 72^\circ\), angle \(C = 72^\circ\), so angle \(B = 180 - 72 - 72 = 36^\circ\). But wait, maybe the equal sides are \(AB\) and \(AC\)? Wait, the ticks are on the two sides from \(A\) to \(B\) and \(A\) to \(C\)? No, the figure shows two ticks on the sides connected to \(B\). So \(BA = BC\), so triangle \(ABC\) with \(BA = BC\), so isosceles with base \(AC\), legs \(BA\) and \(BC\). Then base angles are \(A\) and \(C\), each \(72^\circ\), so vertex angle \(B\) is \(36^\circ\). But the word bank has 36, 72, 108. Wait, maybe I got the equal sides wrong. Suppose the equal sides are \(AB\) and \(AC\), so angles at \(B\) and \(C\) are equal. Then angle \(A = 72^\circ\), so angles \(B\) and \(C = (180 - 72)/2 = 54^\circ\), but that's not in the word bank. Wait, maybe the equal sides are \(AB\) and \(BC\), but angle at \(B\) is the vertex angle. Wait, no, the word bank has 36, 72, 108. Let's check: if angle \(B\) is \(36^\circ\), angle \(A = 72^\circ\), angle \(C = 72^\circ\) (sum 180). If angle \(B\) is \(108^\circ\), then angles \(A\) and \(C = (180 - 108)/2 = 36^\circ\). Ah! That must be it. I had the equal sides wrong. The equal sides are \(AC\) and \(AB\), so angles at \(B\) and \(C\) are equal? No, wait, if the two equal sides are \(AB\) and \(AC\), then angles at \(B\) and \(C\) are equal. But angle at \(A\) is \(72^\circ\), so angles \(B\) and \(C = (180 - 72)/2 = 54^\circ\), not in word bank. Wait, the figure: the two ticks are on the sides from \(B\) to \(A\) and \(B\) to \(C\)? No, maybe the equal sides are \(AB\) and \(AC\), so angle at \(B\) is vertex angle. Wait, angle at \(A\) is \(72^\circ\), sides \(AB\) and \(AC\) are equal, so angles at \(B\) and \(C\) are equal. Then angle \(B = angle C = (180 - 72)/2 = 54\), not in word bank. Wait, the word bank is 36, 72, 108. So let's see: if angle \(B\) is \(108^\circ\), then angles \(A\) and \(C = (180 - 108)/2 = 36^\circ\), but angle \(A\) is given as \(72^\circ\). Wait, no—wait, the angle at \(A\) is \(72^\circ\), and the two equal sides are \(AB\) and \(BC\), so angle at \(A\) (opposite \(BC\)) and angle at \(C\) (opposite \(AB\)) are equal. So angle \(A = angle C = 72^\circ\), so angle \(B = 180 - 72 - 72 = 36^\circ\). But the word bank has 36, 72, 108. So maybe blank 1 is angle \(B\) is 36, blank 2? Wait, maybe the triangle is isosceles with \(AB = BC\), so angle \(A = angle C = 72^\circ\), angle \(B = 36^\circ\). So blank 1 (maybe angle \(B\)) is 36, blank 2 (maybe another angle? No, angle \(A\) is 72, angle \(C\) is 72, angle \(B\) is 36. Wait, the word bank has 36, 72, 108. So perhaps the first blank is \(m\angle B = 36^\circ\), second blank? Wait, maybe the sides: if it's an isosceles triangle with angles 72, 72, 36, or 36, 36, 108. Wait, angle at \(A\) is 72, so if the equal angles are 72, then the third angle is 36. So for blank 1 (maybe \(m\angle B\)): 36, blank 2 (maybe \(m\angle C\)): 72? Wait, angle \(A\) is 72, angle \(C\) is 72, so blank 2 is 72. So:

Blank 1: 36 (m∠B), Blank 2: 72 (m∠C) or vice versa. But let's confirm:

In triangle \(ABC\), \(AB = BC\) (ticks), so ∠A = ∠C (angles opposite equal sides). ∠A = 72°, so ∠C = 72°. Then ∠B = 180 - 72 - 72 = 36°. So:

Blank 1 (e.g., \(m\angle B\)): 36

Blank 2 (e.g., \(m\angle C\)): 72

Answer:

Blank 1: 36, Blank 2: 72 (depending on the statements, but based on angle calculations, \(m\angle B = 36^\circ\), \(m\angle C = 72^\circ\) or \(m\angle A = 72^\circ\), \(m\angle B = 36^\circ\))