Solution: The measures of two angles of a triangle are

Concept Check Use the given figure to find the measures of the numbered angles, given that lines m and n are parallel.

In the figure here, if the measure of one of the angles is known, explain how the remaining measures can be found.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

Find the measure of each marked angle. In Exercises 1114, m and n are parallel. See Examples 1 and 2.

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 37, 52

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 29, 104

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 147 12_, 30 19_

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 136 50_, 41 38_

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 74.2, 80.4

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 29.6, 49.7

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 51_ 20_ 14_, 106_ 10_ 12_

The measures of two angles of a triangle are given. Find the measure of the third angle. See Example 2. 17_ 41_ 13_, 96_ 12_ 10_

Can a triangle have angles of measures 85 and 100? Explain.

Can a triangle have two obtuse angles? Explain.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Concept Check Classify each triangle in Exercises 2536 as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Write a definition of isosceles right triangle.

Explain why the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Use this figure to discuss why the measures of the angles of a triangle must add up to the same sum as the measure of a straight angle.

Carpentry Technique The following technique is used by carpenters to draw a 60 angle with a straightedge and a pair of compasses. Explain why this technique works. (Source: Hamilton, J. E. and M. S. Hamilton, Math to Build On, Construction Trades Press.) Draw a straight line segment, and mark a point near the midpoint. Now place the tip on the marked point, and draw a semicircle. Without changing the setting of the pair of compasses, place the tip at the right intersection of the line and the semicircle, and then mark a small arc across the semicircle. Finally, draw a line segment from the marked point on the original segment to the point where the arc crosses the semicircle. This will form a 60 angle with the original segment.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Concept Check Name the corresponding angles and the corresponding sides of each pair of similar triangles. See the discussion preceding Example 3.

Find the unknown side lengths labeled with a variable in each pair of similar triangles. See Example 4.

Find the unknown side lengths labeled with a variable in each pair of similar triangles. See Example 4.

Find the unknown side lengths labeled with a variable in each pair of similar triangles. See Example 4.

Find the unknown side lengths labeled with a variable in each pair of similar triangles. See Example 4.

Find the unknown side lengths labeled with a variable in each pair of similar triangles. See Example 4.

Find the unknown side lengths labeled with a variable in each pair of similar triangles. See Example 4.

Height of a Tree A tree casts a shadow 45 m long. At the same time, the shadow cast by a vertical 2-m stick is 3 m long. Find the height of the tree.

Height of a Lookout Tower A forest fire lookout tower casts a shadow 180 ft long at the same time that the shadow of a 9-ft truck is 15 ft long. Find the height of the tower.

Lengths of Sides of a Triangle On a photograph of a triangular piece of land, the lengths of the three sides are 4 cm, 5 cm, and 7 cm, respectively. The shortest side of the actual piece of land is 400 m long. Find the lengths of the other two sides.

Height of a Lighthouse The Biloxi lighthouse in the figure casts a shadow 28 m long at 7 p.m. At the same time, the shadow of the lighthouse keeper, who is 1.75 m tall, is 3.5 m long. How tall is the lighthouse?

Height of a Building A house is 15 ft tall. Its shadow is 40 ft long at the same time that the shadow of a nearby building is 300 ft long. Find the height of the building.

Height of a Carving of Lincoln Assume that Lincoln was 6 13 ft tall and his head 34 ft long. Knowing that the carved head of Lincoln at Mt. Rushmore is 60 ft tall, find how tall his entire body would be if it were carved into the mountain.

In each diagram, there are two similar triangles. Find the unknown measurement. (Hint: In the sketch for Exercise 63, the side of length 100 in the small triangle corresponds to the side of length 100 + 120 = 220 in the large triangle.)

In each diagram, there are two similar triangles. Find the unknown measurement. (Hint: In the sketch for Exercise 63, the side of length 100 in the small triangle corresponds to the side of length 100 + 120 = 220 in the large triangle.)

In each diagram, there are two similar triangles. Find the unknown measurement. (Hint: In the sketch for Exercise 63, the side of length 100 in the small triangle corresponds to the side of length 100 + 120 = 220 in the large triangle.)

In each diagram, there are two similar triangles. Find the unknown measurement. (Hint: In the sketch for Exercise 63, the side of length 100 in the small triangle corresponds to the side of length 100 + 120 = 220 in the large triangle.)

Solar Eclipse on Earth The sun has a diameter of about 865,000 mi with a maximum distance from Earths surface of about 94,500,000 mi. The moon has a smaller diameter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moons umbra (shadow) to reach the surface of Earth. (Source: Karttunen, H., P. Krger, H. Oja, M. Putannen, and K. Donners, Editors, Fundamental Astronomy, Fourth Edition, Springer-Verlag.) (a) Calculate the maximum distance that the moon can be from Earth and still have a total solar eclipse occur. (Hint: Use similar triangles.) (b) The closest approach of the moon to Earths surface was 225,745 mi and the farthest was 251,978 mi. (Source: World Almanac and Book of Facts.) Can a total solar eclipse occur every time the moon is between Earth and the sun?

Solar Eclipse on Neptune (Refer to Exercise 67) The suns distance from Neptune is approximately 2,800,000,000 mi (2.8 billion mi). The largest moon of Neptune is Triton, with a diameter of approximately 1680 mi. (Source: World Almanac and Book of Facts.) (a) Calculate the maximum distance that Triton can be from Neptune for a total eclipse of the sun to occur on Neptune. (Hint: Use similar triangles.) (b) Triton is approximately 220,000 mi from Neptune. Is it possible for Triton to cause a total eclipse on Neptune?

Solar Eclipse on Mars (Refer to Exercise 67) The suns distance from the surface of Mars is approximately 142,000,000 mi. One of Mars two moons, Phobos, has a maximum diameter of 17.4 mi. (Source: World Almanac and Book of Facts.) (a) Calculate the maximum distance that the moon Phobos can be from Mars for a total eclipse of the sun to occur on Mars. (b) Phobos is approximately 5800 mi from Mars. Is it possible for Phobos to cause a total eclipse on Mars?

Solar Eclipse on Jupiter (Refer to Exercise 67) The suns distance from the surface of Jupiter is approximately 484,000,000 mi. One of Jupiters moons, Ganymede, has a diameter of 3270 mi. (Source: World Almanac and Book of Facts.) (a) Calculate the maximum distance that the moon Ganymede can be from Jupiter for a total eclipse of the sun to occur on Jupiter. (b) Ganymede is approximately 665,000 mi from Jupiter. Is it possible for Ganymede to cause a total eclipse on Jupiter?

Sizes and Distances in the Sky Astronomers use degrees, minutes, and seconds to measure sizes and distances in the sky along an arc from the horizon to the zenith point directly overhead. An adult observer on Earth can judge distances in the sky using his or her hand at arms length. An outstretched hand will be about 20 arc degrees wide from the tip of the thumb to the tip of the little finger. A clenched fist at arms length measures about 10 arc degrees, and a thumb corresponds to about 2 arc degrees. (Source: Levy, D. H., Skywatching, The Nature Company.) (a) The apparent size of the moon is about 31 arc minutes. What part of your thumb would cover the moon? (b) If an outstretched hand plus a fist cover the distance between two bright stars, about how far apart in arc degrees are the stars?

Estimates of Heights There is a relatively simple way to make a reasonable estimate of a vertical height. Hold a 1-ft ruler vertically at arms length as you approach the object to be measured. Stop when one end of the ruler lines up with the top of the object and the other end with its base. Now pace off the distance to the object, taking normal strides. The number of paces will be the approximate height of the object in feet. Furnish the reasons in parts (a)(d), which refer to the figure. (Assume that the length of one pace is EF.) Then answer the question in part (e).