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Core Connections Integrated I, 2013

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Core Connections Integrated I, 2013 View details

2. Section 8.2

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Exercise 122 Page 484

a By the Triangle Sum Theorem, we know that the three angle measures of a triangle sum to

18 0^{\circ} .

With this, we can write an equation.

(3 x - 2^{\circ}) + (x + 8^{\circ}) + (2 x + 3^{\circ}) = 18 0^{\circ}

Let's solve for

x

in this equation.

(3 x - 2) + (x + 8) + (2 x + 3) = 180

Remove parentheses

3 x - 2 + x + 8 + 2 x + 3 = 180

Add and subtract terms

6 x + 9 = 180

$LHS - 9 = RHS - 9$

6 x = 171

$LHS / 6 = RHS / 6$

x = 28.5

b Examining the diagram, we notice that the two angles are alternate exterior angles. Since the two lines cut by the transversal are parallel, we know by the Alternate Exterior Angles Theorem that these angles are congruent. This means we can equate their expressions.

6 x - 2 8^{\circ} = 4 x + 1 8^{\circ}

Let's solve this equation for

x .

6 x - 28 = 4 x + 18

$LHS - 4 x = RHS - 4 x$

2 x - 28 = 18

$LHS + 28 = RHS + 28$

2 x = 46

$LHS / 2 = RHS / 2$

x = 23

c Notice that the triangle is isosceles which means it has two congruent legs. By the Base Angles Theorem, we know that the base angles are congruent. Let's call these angles

y

and add them to the diagram.

Let's isolate a few parts of the diagram.

Here, we see that $y$ and $5 6^{\circ}$ are corresponding angles and since the two lines cut by the transversal are parallel, we know by the Corresponding Angles Theorem that they are congruent. Therefore, we can say that $y = 5 6^{\circ} .$ Let's add this information to the diagram.

Finally, we can by the Triangle Sum Theorem, write an equation including

x .

x + 5 6^{\circ} + 5 6^{\circ} = 18 0^{\circ}

Let's solve this equation for

x .

x + 56 + 56 = 180

Add terms

x + 112 = 180

$LHS - 112 = RHS - 112$

x = 68

Subchapter links

8.2.1 Curve Fitting p.471-472

8.2.2 Curved Best-Fit Models p.478-480

8.2.3 Solving a System of Exponential Functions Graphically p.483-484