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McGraw Hill Integrated II, 2012

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McGraw Hill Integrated II, 2012 View details

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Exercise 1 Page 403

Find the missing angle measures first.

9

Practice makes perfect

In general it is not an easy task to find the length of a side of a triangle if the measure of two sides and the corresponding angle are given. However, this is a special case. The information on the figure guarantees that this is an equilateral triangle. To confirm this, let's start with finding the missing angle measures.

Finding the Missing Angle Measures

Notice that the given lengths of the sides AB and AC indicate that triangle △ ABC is isosceles. According to the Isosceles Triangle Theorem, this implies that ∠ ACB and ∠ ABC are congruent.

Congruent angles have the same measure. m∠ ACB=m∠ ABC Let's use the Triangle Angle-Sum Theorem to find the measure of the missing angles.

m∠ ACB+m∠ ABC+m∠ CAB=180

m∠ ACB= m∠ ABC, m∠ CAB= 60

m∠ ABC+m∠ ABC+ 60=180

▼

Solve for m∠ ABC

Add terms

2m∠ ABC+60=180

LHS-60=RHS-60

2m∠ ABC=120

.LHS /2.=.RHS /2.

m∠ ABC=60

The measuer of angle ∠ ABC is 60. Since m∠ ACB=m∠ ABC, this also implies that the measure of angle ∠ ACB is also 60.

Finding the Missing Side Length

We now know that the angles of triangle △ ABC have the same measure, so △ ABC is equiangular. According to the converse of the Isosceles Triangle Theorem, this means that triangle △ ABC is equilateral. Since the measures of the two sides are given as 9 units, the third side also has the same length. BC=9

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