Geometry: An Interactive Journey to Mastery

Example 1
Prove that the interior angle of a triangle of largest measure lies opposite the longest side of the triangle.
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Suppose that a triangle has interior angles of measures x and y and
sides of lengths a and b, as shown in Figure 15.1. Assume that b! a.
'UDZDVHJPHQWRIOHQJWKa on the side of length b to create an
isosceles triangle. Mark the congruent base angles of the isosceles
triangle as x 1 , and mark the angle x 2 , as shown in Figure 15.2.
Clearly, x! x 1.
D௘ͽ 3URYHWKDWx 1! y.
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of the triangle.
Solution
D௘ͽ :HVHHDWULDQJOHLQWKHGLDJUDPZLWKLQWHULRUDQJOHVx 2 , yDQGíx 1. Because the angles in a
triangle sum to 180°, we have x 2 + yíx 1 = 180. This gives x 1 = y + x 2 , which shows that x 1! y.
E௘ͽ :HKDYHx! x 1 and x 1! y. It follows that x! y.
F௘ͽ :HKDYHMXVWSURYHGWKDWLIb! a, then the interior angle opposite b is larger than the interior angle
opposite a. If b is the largest side length of the triangle, then the angle opposite b is also larger than the
WKLUGLQWHULRUDQJOHͼ௘E\WKHVDPHUHDVRQLQJ௘ͽ7KXVWKHDQJOHRSSRVLWHb is the largest interior angle.
Example 2
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E௘ͽ *LYHDQH[DPSOHRIDQREWXVHLVRVFHOHVWULDQJOHDQGSURYHWKDW\RXUWULDQJOHUHDOO\LVREWXVH
Also give an example of a right isosceles triangle.
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a b

x y

Figure 15.1

a

a

x x > x y

1

x 1

x 1

x 2

Figure 15.2