Question

are these shapes congruent?
k
30°
56 ft
56 ft
75°
75°
29 ft
j
o
29 ft
75°
p
75°
56 ft
56 ft
30°
q
yes
no

Explanation:

Step1: Analyze Triangle KIJ

In triangle \( KIJ \), two sides are \( 56 \) ft, so it's isosceles. The vertex angle at \( K \) is \( 30^\circ \), so the base angles at \( I \) and \( J \) are calculated as \( \frac{180^\circ - 30^\circ}{2} = 75^\circ \). The base \( IJ \) is \( 29 \) ft, and the equal sides \( KI = KJ = 56 \) ft.

Step2: Analyze Triangle OPQ

In triangle \( OPQ \), we check angles and sides. Angles at \( P \) and \( O \) are \( 75^\circ \), angle at \( Q \) is \( 30^\circ \) (since \( 180 - 75 - 75 = 30 \)). Side \( OQ = 56 \) ft, \( PQ = 56 \) ft, and \( OP = 29 \) ft.

Step3: Check Congruence Conditions

Using the Side - Angle - Side (SAS) or Angle - Side - Angle (ASA) criteria:

• For triangle \( KIJ \) and \( OPQ \), we can match sides and angles:
• \( KI = PQ = 56 \) ft, \( KJ = OQ = 56 \) ft, \( IJ = OP = 29 \) ft.
• Angles between the sides: In \( KIJ \), the angle between \( KI \) and \( KJ \) is \( 30^\circ \), in \( OPQ \), the angle between \( PQ \) and \( OQ \) (corresponding sides) is \( 30^\circ \) (angle at \( Q \) and angle at \( K \) are equal). Also, the base angles of \( 75^\circ \) match. So by SAS (or ASA) congruence criterion, the triangles are congruent.

Answer:

yes