Question

in parts (a)-(g), determine if the following pairs of triangles must be congruent. if they are congruent, write a congruence statement and the theorem you used.
the following diagrams below are not necessarily drawn to scale.

Explanation:

Step1: Check congruence for part (a)

• In \(\triangle ABC\) and \(\triangle DEF\), we have \(AB = DE = 8\), \(BC=DF = 11\), and \(\angle B=\angle E = 70^{\circ}\).
• By the SAS (Side - Angle - Side) congruence theorem, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
• So, \(\triangle ABC\cong\triangle DEF\) (SAS).

Step2: Check congruence for part (b)

• In \(\triangle W L J\) and \(\triangle V T P\), we have \(LJ = TP=12\), \(WJ = VP = 16\), but the angle in \(\triangle W L J\) is not the included angle between the sides \(LJ\) and \(WJ\), while in \(\triangle V T P\) the angle is between the sides \(TP\) and \(VP\).
• There is no congruence theorem (SSS, SAS, AAS, ASA, HL) that can be applied here. So the triangles are not congruent.

Step3: Check congruence for part (c)

• In \(\triangle ABD\) and \(\triangle CBD\), we have \(BD = BD\) (common side), \(\angle ADB=\angle CDB = 90^{\circ}\), and \(AB = CB\) (given).
• By the HL (Hypotenuse - Leg) congruence theorem (applicable for right - angled triangles), if the hypotenuse and one leg of a right - angled triangle are equal to the hypotenuse and one leg of another right - angled triangle, the triangles are congruent.
• So, \(\triangle ABD\cong\triangle CBD\) (HL).

Step4: Check congruence for part (d)

• In \(\triangle ABC\) and \(\triangle DBC\), we have \(AB = DC\), \(BC=BC\) (common side), and \(AC = DB\).
• By the SSS (Side - Side - Side) congruence theorem, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
• So, \(\triangle ABC\cong\triangle DCB\) (SSS).

Answer:

a. Yes, SAS, \(\triangle ABC\cong\triangle DEF\)
b. No
c. Yes, HL, \(\triangle ABD\cong\triangle CBD\)
d. Yes, SSS, \(\triangle ABC\cong\triangle DCB\)