Download GEOMETRY HOMEWORK EXAMPLES – SECTION 5

GEOMETRY HOMEWORK EXAMPLES – SECTION 5-5 Inequalities Involving Two Triangles
#5 page 270
Notice that the top triangle is an equilateral triangle. Therefore, all the angles in that triangle are 60
degrees. In the bottom triangle, since we know that it is isosceles and the top angle is 45 degrees, we
can just subtract & divide by two to find the other two angles (180 – 45)/2 = 67.5 degrees each. Label
the triangle. I did my work below:
60°
x+5
60°
45°
60°
We can use the SAS Inequality theorem to conclude that, since two sides of
the top triangle are congruent to two sides of the bottom triangle and the
included angle 45° is less than 60°, the side across from 45 ° "3x - 7" must be
less than the side across from 60° "x + 5". We get:
67.5°
67.5° 3x-7
45°<60°
3x - 7 < x + 5
2x < 12
x<6
Remember that side lengths also need to be positive. Thus, each side must
be greater than zero. Take the smallest side length to ensure this.
3x - 7 > 0
3x > 7
x > 7/3
Putting the two inequalities together into a compound inequality, we get
7/3 < x < 6
#17 page 270
Notice that side all the sides of the left triangle are “x + 2” units, the left triangle must be an equilateral
triangle. Therefore, we can label all the angles in the left triangle as 60 degrees. I did my work below:
We can use the SAS Inequality theorem to conclude that, since two sides of
the left triangle are congruent to two sides of the right triangle and the
included angle 58° is less than 60°, the side across from 58 ° "2x - 8" must be
less than the side across from 60° "x + 2". We get:
x+2
60°
60° 58°
x+2
58°<60°
2x - 8 < x + 2
x+2
x < 10
x+2
60°
2x - 8
Remember that side lengths also need to be positive. Thus, each side must
be greater than zero. Take the smallest side length to ensure this.
2x - 8 > 0
2x > 8
x>4
Putting the two inequalities together into a compound inequality, we get
the 4 < x < 10