THIRD ANGLES THEOREM

Third Angles Theorem :

This theorem tells us angles congruence between triangles.

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

In the diagram given below, if ∠A ∠D and∠B ∠E, then ∠C ∠F.

Third angles theorem - Problems

Problem 1 :

In the diagram given below, ∠L  ≅  ∠P and ∠M  ≅  ∠Q, find m∠N and m∠R. Check whether ∠N ≅ ∠R and justify your answer.

Solution :

Given : ∠L  ≅  ∠P and ∠M  ≅  ∠Q

 From the figure above, we have m∠L  =  105°Because ∠L  ≅  ∠P, we have m∠P  =  105° From the figure above, we have m∠Q  =  45°Because ∠Q  ≅  ∠M, we have m∠M  =  45°

ΔLMN :

By the Triangle Sum theorem, we have

m∠L + m∠M + m∠N  =  180°

Substitute 105° for m∠L and 45° for m∠M.

105° + 45° + m∠N  =  180°

Simplify.

150° + m∠N  =  180°

Subtract 150° from both sides.

m∠N  =  30°

ΔPQR :

By the Triangle Sum theorem, we have

m∠P + m∠Q + m∠R  =  180°

Substitute 105° for m∠P and 45° for m∠Q.

105° + 45° + m∠R  =  180°

Simplify.

150° + m∠R  =  180°

Subtract 150° from both sides.

m∠R  =  30°

Because m∠N = 30° and m∠R = 30°, we have

m∠N  =  m∠R ------> ∠N ≅ ∠R

Justification :

By the Third Angles Theorem, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

So, ∠N ≅ ∠R

Problem 2 :

In the diagram given below, ∠B  ≅  ∠U and ∠C  ≅  ∠V, find m∠A and using the Third Angles Theorem to find m∠T.

Solution :

Given : ∠B  ≅  ∠U and ∠C  ≅  ∠V

From the figure, we have m∠U  =  59°

Because ∠B  ≅  ∠U, we have m∠B  =  59°

In ΔABC, by the Triangle Sum theorem, we have

m∠A + m∠B + m∠C  =  180°

Substitute 59° for m∠B and 55° for m∠C.

m∠A + 59° + 55°  =  180°

Simplify.

m∠A + 114°  =  180°

Subtract 114° from both sides.

m∠A  =  66°

By the Third Angles Theorem, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

So, we have

∠T ≅ ∠A

m∠T = ∠A

m∠T =  66°

Problem 3 :

Find the value of x in the diagram given below.

Solution :

In the diagram given above, ∠N ≅ ∠R and ∠L ≅ ∠S. From the Third angles theorem, we know that ∠M ≅ ∠T. So, m∠M  =  m∠T.

From the triangle sum theorem, we have

m∠L + m∠M + m∠N  =  180°

65° + 55° + m∠M  =  180°

Simplify

120° + m∠M  =  180°

Subtract 120° from both sides.

m∠M  =  60°

By the theorem, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

So, we have

∠M ≅ ∠T ------> m∠M  =  m∠T

Substitute 60° for m∠M and (2x + 30)° for m∠M.

60°  =  (2x + 30)°

60  =  2x + 30

Subtract 30 from both sides.

30  =  2x

Divide both sides by 2.

15  =  x

Problem 4 :

Decide whether the triangles are congruent. Justify your reasoning.

Solution :

From the diagram, we are given that all three pairs of corresponding sides are congruent.

RP ≅ MN, PQ ≅ NQ and Q≅ QM

Because ∠P and ∠N have the same measure, ∠P ≅ ∠N.

By the Vertical Angles Theorem, we know that

∠PQR  ≅  MQN

In ΔPQR and ΔMQN, ∠P ≅ ∠N and ∠PQR  ≅  MQN.

By the theorem, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

So, we have

R  ≅  M

So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent.

By the definition of congruent angles,

ΔPQR  ≅  ΔNQM

After having gone through the stuff given above, we hope that the students would have understood the theorem on third angles.

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