THIRD ANGLE THEOREM

Example 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

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

Example 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°

Example 3 : 

Find the value of x in the diagram given below. 

Solution : 

In the diagram 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

Example 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 QR ≅ 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

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