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Comparing Angles and Sides in Triangles

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Comparing Angles and Sides in Triangles

What if you were told that a triangle has sides that measure 3, 4, and 5? How could you determine which of the triangle's angles is largest? Smallest? After completing this Concept, you'll be able to use triangle theorems to solve problems like this one.

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CK-12 Comparing Angles & Sides in Triangles

James Sousa: Proof that the Angle of a Triangle Opposite The Longest Side is the Largest Angle

Guidance

Look at the triangle below. The sides of the triangle are given. Can you determine which angle is the largest? The largest angle will be opposite 18 because that is the longest side. Similarly, the smallest angle will be opposite 7, which is the shortest side.

This idea is actually a theorem: If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.

The converse is also true: If one angle in a triangle is larger than another angle in that triangle, then the side opposite the larger angle will be longer than the side opposite the smaller angle.

We can extend this idea into two theorems that help us compare sides and angles in two triangles If we have two congruent triangles \triangle ABC and \triangle DEF, marked below:

Therefore, if AB = DE, BC = EF, and m \angle B = m \angle E, then AC = DF.

Now, let’s make m \angle B > m \angle E. Would that make AC > DF? Yes. This idea is called the SAS Inequality Theorem.

The SAS Inequality Theorem: If two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then the third side of the first triangle is longer than the third side of the second triangle.

If \overline{AB} \cong \overline{DE}, \overline{BC} \cong \overline{EF} and m \angle B > m \angle E, then \overline{AC} > \overline{DF}.

If we know the third sides as opposed to the angles, the opposite idea is also true and is called the SSS Inequality Theorem.

SSS Inequality Theorem: If two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle's two congruent sides is greater in measure than the included angle of the second triangle's two congruent sides.

If \overline{AB} \cong \overline{DE}, \ \overline{BC} \cong \overline{EF} and \overline{AC} > \overline{DF}, then m \angle B > m \angle E.

Example A

List the sides in order, from shortest to longest.

First, find m \angle A. From the Triangle Sum Theorem:

m \angle A + 86^\circ + 27^\circ &= 180^\circ\\m \angle A &= 67^\circ

86^\circ is the largest angle, so AC is the longest side. The next angle is 67^\circ, so BC would be the next longest side. 27^\circ is the smallest angle, so AB is the shortest side. In order, the answer is: AB, \ BC, \ AC.

Example B

List the angles in order, from largest to smallest.

Just like with the sides, the largest angle is opposite the longest side. The longest side is BC, so the largest angle is \angle A. Next would be \angle B and then \angle C.

Example C

List the sides in order, from least to greatest.

To solve, let’s start with \triangle DCE. The missing angle is 55^\circ. By the theorem presented in this Concept, the sides, in order from least to greatest are CE, \ CD, and DE.

For \triangle BCD, the missing angle is 43^\circ. Again, by the theorem presented in this Concept, the order of the sides from least to greatest is BD, \ CD, and BC.

By the SAS Inequality Theorem, we know that BC > DE, so the order of all the sides would be: BD, \ CE, \ CD, \ DE, \ BC.

CK-12 Comparing Angles & Sides in Triangles

Guided Practice

1. If \overline{XM} is a median of \triangle XYZ and XY > XZ, what can we say about m \angle 1 and m \angle 2?

2. List the sides of the two triangles in order, from shortest to longest.

3. Below is isosceles triangle \triangle ABC. List everything you can about the sides and angles of the triangle and why.

Answers:

1. M is the midpoint of \overline{YZ}, so YM = MZ. MX = MX by the Reflexive Property and we know XY > XZ.

We can use the SSS Inequality Theorem Converse to say m \angle 1 > m \angle 2.

2. There are no congruent sides or angles. Look at each triangle separately.

\triangle XYZ: The missing angle is 42^\circ. By the theorem presented in this lesson, the order of the sides from shortest to longest is YZ, \ XY, and XZ.

\triangle WXZ: The missing angle is 55^\circ. The order of the sides from shortest to longest is XZ, \ WZ, and WX.

Because the longest side in \triangle XYZ is the shortest side in \triangle WXZ, we can put all the sides together in one list: YZ, \ XY, \ XZ, \ WZ, \ WX.

3. AB = BC because it is given.

m \angle A = m \angle C because if sides are equal than their opposite angles must be equal..

AD < DC because m \angle ABD < m \angle CBD and because of the SAS Triangle Inequality Theorem.

Practice

For questions 1-3, list the sides in order from shortest to longest.

For questions 4-6, list the angles from largest to smallest.

  1. Draw a triangle with sides 3 cm, 4 cm, and 5 cm. The angle measures are 90^\circ, \ 53^\circ, and 37^\circ. Place the angle measures in the appropriate spots.
  2. Draw a triangle with angle measures 56^\circ, \ 54^\circ and the included side is 8 cm. What is the longest side of this triangle?
  3. Draw a triangle with sides 6 cm, 7 cm, and 8 cm. The angle measures are 75.5^\circ, \ 58^\circ, and 46.5^\circ. Place the angle measures in the appropriate spots.
  4. What conclusions can you draw about x?
  5. Compare m \angle 1 and m \angle 2.
  6. List the sides from shortest to longest.
  7. Compare m \angle 1 and m \angle 2. What can you say about m \angle 3 and m \angle 4?

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