We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.

+0

# Triangle Angles and Congruent Triangles

0
548
2

In the figure, $$AD=CD$$ and $$AB=CB$$.

(i) Prove that $$\overline{BD}$$ bisects angle $$ADC$$ (i.e. that $$\overline{BD}$$ cuts angle $$ADC$$ into two equal angles).

(ii) Prove that $$\overline{AC}$$ and $$\overline{DB}$$ are perpendicular. Aug 17, 2017

### 2+0 Answers

#1
+1

AD is congruent to CD (given)

AB is congruent to CB (given)

DB is contruent to DB (identity)

Therefore, triangle(ADB) is congruent to triangle(DCB) (side-side-side)

and angle(ADB) is congruent to angle(CDB) (corresponding parts of congruent triangles are congruent)

Since angle(ADB) is congruent to angle(CDG), BD bisects angle(ADC).

AD is congruent to CD (given)

angle(ADB) is congruent to angleCDB) (proven above)

Therefore, triangle(ADE) is congruent to triangle(CDE) (side-angle-side)

and angle(AED) is congruent to angle(CED) (corresponding parts of congruent triangles are congruent)

Since angle(AED) and angle(CED) formf a linear pair (a straight line), angle(AED) and angle(CED) are right angles.

Thus, AC and DB are perpendicular.

Aug 17, 2017
#2
+1

Here's another method to prove condition ii ($$\overline{AC}\perp\overline{DB}$$). I will utilize a two-column proof:

 $$AD=CD$$ $$AB=CB$$ Given $$\overline{AD}\cong\overline{CD}$$ $$\overline{AB}\cong\overline{CB}$$ Definition of congruent segments Figure $$ABCD$$ is a kite Definition of a kite (If a quadrilateral has two unique pairs of sides that are congruent, then the figure is a kite) $$\overline{AC}\perp\overline{DB}$$ Property of a kite (Diagonals of a kite are perpendicular)

Aug 17, 2017