Question

for each figure below, determine (by using the markings) if there is a pair of congruent triangles. if there is, name the congruence and give the property justifying the congruence. note that the pairs of triangles are drawn as congruent, but you should not rely on how they are drawn in determining your answers. if necessary, you may learn what the markings on a figure indicate. (a) m o n p q r not necessarily congruent congruent: △ mno ≅ △□ by the select (b) w u y x not necessarily congruent congruent: △ uvw ≅ △□ by the select (c) h g i l j k not necessarily congruent congruent: △ ghi ≅ △□ by the select

Explanation:

(a)

Step1: Identify corresponding parts

In \(\triangle MNO\) and \(\triangle PQR\), \(\angle M=\angle P = 90^{\circ}\), \(MO = PQ\) (marked with one tick), \(MN=PR\) (marked with one tick).

Step2: Apply congruence property

By the Hypotenuse - Leg (HL) congruence property (for right - angled triangles, if the hypotenuse and one leg are equal).

(b)

Step1: Identify vertical angles

\(\angle UVW\) and \(\angle YVX\) are vertical angles, so \(\angle UVW=\angle YVX\). Also, \(UW = XY\) (marked with one tick), \(VW=VX\) (marked with one tick).

Step2: Apply congruence property

By the Side - Angle - Side (SAS) congruence property (two sides and the included angle are equal).

(c)

Step1: Check side - side - side

\(GI\) and \(JK\) are not necessarily equal. \(GH = LJ\) (marked with one tick), \(HI = LK\) (marked with one tick), but we don't have information about the third side.

Answer:

(a) Congruent: \(\triangle MNO\cong\triangle PQR\) by the HL (Hypotenuse - Leg) property.
(b) Congruent: \(\triangle UVW\cong\triangle YVX\) by the SAS (Side - Angle - Side) property.
(c) Not Necessarily Congruent.