Question

35 multiple choice 2.5 points
determine if the triangles are similar. if so, write the postulate (sss, sas or aa). if they are not similar, write na.
image of two right triangles
options: sas, na, sss, aa
36 multiple choice 2.5 points
quadrilateral abcd is a rhombus. find the value of y if m∠1 = 2y + 6.
image of rhombus abcd with diagonals intersecting
options: 87, 48, 42, 90

Explanation:

Question 35

Step1: Analyze triangle angles

Triangle \( XYZ \) has a right angle (\( \angle Y = 90^\circ \)) and \( \angle Z = 47^\circ \), so the third angle \( \angle X = 180^\circ - 90^\circ - 47^\circ = 43^\circ \).
Triangle \( ABC \) has a right angle (\( \angle B = 90^\circ \)) and \( \angle C = 43^\circ \), so \( \angle A = 180^\circ - 90^\circ - 43^\circ = 47^\circ \).

Step2: Apply AA similarity

Two angles of \( \triangle XYZ \) ( \( 90^\circ \) and \( 43^\circ \) ) match two angles of \( \triangle ABC \) ( \( 90^\circ \) and \( 43^\circ \) ). By the AA (Angle - Angle) similarity postulate, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Step1: Recall rhombus diagonals property

In a rhombus, the diagonals are perpendicular bisectors of each other, so \( \angle 1 = 90^\circ \) (diagonals intersect at right angles).

Step2: Solve for \( y \)

Given \( m\angle 1 = 2y + 6 \) and \( \angle 1 = 90^\circ \), set up the equation:
\( 2y + 6 = 90 \)
Subtract 6 from both sides: \( 2y = 90 - 6 = 84 \)
Divide by 2: \( y = \frac{84}{2} = 42 \)

Answer:

AA

Question 36