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

Example 5 :

The triangle are similar, find the value of x.

Solution :

Since the above triangles are similar, corresponding angles will be equal and measures of corresponding sides will be in the same ratio.

x = 50

Example 6 :

In the diagram below triangles ABC and QLP are similar.

a) Which side can you find in triangle ABC? Find that side.

b) Find the measure of ∠Q in the triangle.

c) Use the angles in triangle QLP to find all the missing angles in triangle ABC.

d) Extra: Write a ratio that would be equivalent to PL/PQ.

Solution :

a) Since the triangles are similar,

AB and QL are corresponding sides.

• <B and <L are corresponding angles.
• <A and <Q are corresponding angles.
• <C and <P are corresponding angles.

a)

AB/QL = AC/QP = BC/PL

AC/24 = 30/40

AC/24 = 3/4

AC = (3/4) x 24

AC = 18

b) In triangle QLP,

<P + <Q + <L = 180

110 + <Q + 35 = 180

<Q + 145 = 180

<Q = 180 - 145

<Q = 35

c) In triangle ABC, <B = 35 and <A = 35

d) PL/PQ = 40/24

= 5/3

Example 6 :

a) Find the missing angle in each triangle. How does this show that the triangles are similar?

b) If WI = 9.4 and WL  = 10. Find the side IL using the Pythagorean Theorem (to one decimal)

c) If the scale factor from triangle WIL to triangle ATC is ½, find all the missing sides of triangle TAC.

Solution :

In triangle WIL,

<W + <I + <L = 180

<W + 90 + 70 = 180

<W + 160 = 180

<W = 180 - 160

<W = 20

Since the triangles are equiangular, they are similar.

b)

WL² = WI² + IL²

10² = 9.4² + IL²

100 - 88.36 = IL²

IL² = 11.64

IL = 3.4

c) WL/TC = IL/AT = WI/AC

10 / TC = 3.4/AT = 9.4/AC = 1/2

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