Proof. Consider and with . We must show that .
Construct so that . On , take point so that . Either is on or is not on . In either case, we must have by SAS postulate and by CPCTC.
Case 1: is on
By the Segment Addition Postulate , so . But from our congruence above we had . By substitution we have and we have proven case 1.
Case 2: is not on .
Construct the bisector of so that it intersects at point . Draw and .
Recall that .
Note that by SAS postulate. Then .
So, by the Triangle Inequality Theorem.
Now by the segment addition postulate, and by our original construction of , so by substitution we have or and we have proven case 2.
We can also prove the converse of the Hinge theorem.
SSS Inequality Theorem-Converse of Hinge 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 is greater in measure than the included angle of the second triangle.
Proof. In order to prove the theorem, we will again use indirect reasoning as we did in proving Theorem 5-11.
Consider and with We must show that .
1. Assume that is not greater than . Then either or .
Case 1: If , then and are congruent by SAS postulate and we have . But this contradicts the given condition that .
2. Case 2: If , then by Theorem 5-11. This contradicts the given condition that .
3. Since we get contradictions in both cases, then our original assumption was incorrect and we must have .
We can now look at some problems that we could solve with these theorems.
Example 1
What we can deduce from the following diagrams.
1. Given: is a median of with .
Since and then Theorem 5-14 applies and we have .
2. Given: as indicated.
Since we have two sides of congruent with two sides of , then Theorem 5-13 applies and we have .
Lesson Summary
In this lesson we:
Stated and proved theorems that helped determine relationships among the angles and sides of a pair of triangle.
Applied the SAS and SSS Inequality Theorems to solve problems.
Review Questions
Use the theorems to make deductions in problems 1-5. List any theorems or postulates you use.
Suppose that is acute and is obtuse
In problems 6-10, determine whether the assertion is true and give reasons to support your answers.
Assertion: and .
Assertion: in the figure below.
Assertion: .
In problems 9-10, is the assertion true or false?
Assertion: .
Consider is a right triangle with median from as indicated. Assertion: .
Review Answers
by Theorem 5-13.
by Theorem 5-13.
by Theorem 5-14.
We cannot deduce anything as we know nothing about the included angle nor the third side of each triangle.
by Theorem 5-13.
The assertions are true. by Theorem 5-14. Since both triangles are isosceles and , then an implication of the fact that base angles are congruent will imply that .
The assertion is false. The two triangles have two sides congruent. The measure of angle (adjacent to the angle) is since the triangle is equilateral. Hence, Theorem 5-13 applies and so .
The assertion is true by Theorem 5-14: .
The assertion is false. Theorem 5-14 applies and we have .
The assertion is false. We do not have enough information to apply the theorems in this example.
Indirect Proof
Learning Objective
Reason indirectly to develop proofs of statement
Introduction
Recall that in proving Theorems about the relationship between the sides and angles of triangle we used a method of proof in which we temporarily assumed that the conclusions was false and then reached a contradiction of the given statements. This method of proving something is called indirect proof. In this lesson we will practice using indirect proofs with both algebraic and geometric examples.
Indirect Proofs in Algebra
Let’s begin our discussion with an algebraic example that we will put into if-then form.
Example 1
If , then
Proof. Let’s assume temporarily that the . Then we can reach a contradiction by applying our standard algebraic properties of real numbers and equations as follows:
This last statement contradicts the given statement that Hence, our assumption is incorrect and we must have .
We can also employ this kind of reasoning in geometric situations. Consider the following theorem which we have previously proven using the Corresponding Angles Postulate:
Theorem: If parallel lines are cut by a transversal, then alternate interior angles are congruent.
Proof. It suffices to prove the theorem for one pair of alternate interior angles. So consider and . We need to show that .
Assume that we have parallel lines and that . We know that lines are parallel, so we have by postulate that corresponding angles are congruent and . Since vertical angles are congruent, we have . So by substitution, we must have , which is a contradiction.
Lesson Summary
In this lesson we:
Illustrated some examples of proof by indirect reasoning, from algebra and geometry.
Points to Consider
Indirect reasoning can be a powerful tool in proofs. In the section on logical reasoning we saw that if there are two possibilities for a statement (such as TRUE or FALSE), if we can show one of them is not true (i.e. show that a statement is NOT FALSE), then the opposite possibility is all we have left (i.e. the statement is TRUE).
Review Questions
Generate a proof by contradiction for each of the following statements.
If is an integer and is even, then is even.
If in we have , then is not equilateral.
If , then .
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
If one angle of a triangle is larger than another angle of a triangle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
The base angles of an isosceles triangle are congruent.
If is an integer and is odd, then is odd.
If we have with , then is not a right angle.
If two angles of a triangle are not congruent, then the sides opposite those angles are not congruent.
Consider the triangle the following figure with , and . Prove that does not bisect .
Review Answers
Assume is odd. Then for some integer , and which is odd. This contradicts the given statement that is even.
Assume is equilateral. Then by definition, the sides are congruent. By the parallel postulate, we can construct a line parallel to the base through point as follows:
From this we can show with alternate interior angles that the triangle is equiangular so . This contradicts the given statement that .
Assume that is not greater than . Then either in which case , which is a contradiction, or , in which case we can solve the quadratic inequality to get , also a contradiction of the fact that .
Assume that the lines are not parallel. Then . But for vertical angles, so we have . This is a contradiction of the fact that .
Hint: This is theorem 5-11.
Assume , say . By Theorem 5-11 we have , which contradicts the fact that we have an isosceles triangle.
Proof follows the lesson example closely. Assume is even. Then it can be shown that must be even, which is a contradiction.
In we have . Assume that is a right angle. Hence, by substitution we have so that , which is a contradiction.
Suppose we have a trian
gle in which the sides opposite two angles are congruent. Then it follows that the triangle must be isosceles. By Isosceles Triangle Theorem, the opposite angles are congruent, which is a contradiction.
Assume does bisect . Hence, the two triangles are congruent by SAS and by CPCTC, which is a contradiction.
Chapter 6: Quadrilaterals
Interior Angles
Learning Objectives
Identify the interior angles of convex polygons.
Find the sums of interior angles in convex polygons.
Identify the special properties of interior angles in convex quadrilaterals.
Introduction
By this point, you have studied the basics of geometry and you’ve spent some time working with triangles. Now you will begin to see some ways to apply your geometric knowledge to other polygons. This chapter focuses on quadrilaterals—polygons with four sides.
Note: Throughout this chapter, any time we talk about polygons, we will assume that we are talking about convex polygons.
Interior Angles in Convex Polygons
The interior angles are the angles on the inside of a polygon.
As you can see in the image, a polygon has the same number of interior angles as it does sides.
Summing Interior Angles in Convex Polygons
You have already learned the Triangle Sum Theorem. It states that the sum of the measures of the interior angles in a triangle will always be . What about other polygons? Do they have a similar rule?
We can use the triangle sum theorem to find the sum of the measures of the angles for any polygon. The first step is to cut the polygon into triangles by drawing diagonals from one vertex. When doing this you must make sure none of the triangles overlap.
Notice that the hexagon above is divided into four triangles.
Since each triangle has internal angles that sum to , you can find out the sum of the interior angles in the hexagon. The measure of each angle in the hexagon is a sum of angles from the triangles. Since none of the triangles overlap, we can obtain the TOTAL measure of interior angles in the hexagon by summing all of the triangles' interior angles. Or, multiply the number of triangles by :
The sum of the interior angles in the hexagon is .
Example 1
What is the sum of the interior angles in the polygon below?
The shape in the diagram is an octagon. Draw triangles on the interior using the same process.
The octagon can be divided into six triangles. So, the sum of the internal angles will be equal to the sum of the angles in the six triangles.
So, the sum of the interior angles is .
What you may have noticed from these examples is that for any polygon, the number of triangles you can draw will be two less than the number of sides (or the number of vertices). So, you can create an expression for the sum of the interior angles of any polygon using for the number of sides on the polygon.
The sum of the interior angles of a polygon with sides is
Example 2
What is the sum of the interior angles of a nonagon?
To find the sum of the interior angles in a nonagon, use the expression above. Remember that a nonagon has nine sides, so will be equal to nine.
So, the sum of the interior angles in a nonagon is .
Interior Angles in Quadrilaterals
A quadrilateral is a polygon with four sides, so you can find out the sum of the interior angles of a convex quadrilateral using our formula.
Example 3
What is the sum of the interior angles in a quadrilateral?
Use the expression to find the value of the interior angles in a quadrilateral. Since a quadrilateral has four sides, the value of will be .
So, the sum of the measures of the interior angles in a quadrilateral is .
This will be true for any type of convex quadrilateral. You’ll explore more types later in this chapter, but they will all have interior angles that sum to . Similarly, you can divide any quadrilateral into two triangles. This will be helpful for many different types of proofs as well.
Lesson Summary
In this lesson, we explored interior angles in polygons. Specifically, we have learned:
How to identify the interior angles of convex polygons.
How to find the sums of interior angles in convex polygons.
How to identify the special properties of interior angles in convex quadrilaterals.
Understanding the angles formed on the inside of polygons is one of the first steps to understanding shapes and figures. Think about how you can apply what you have learned to different problems as you approach them.
Review Questions
Copy the polygon below and show how it can be divided into triangles from one vertex.
Using the triangle sum theorem, what is the sum of the interior angles in this pentagon?
3-4: Find the sum of the interior angles of each polygon below.
Number of sides =
Sum of interior angles =
Number of sides =
Sum of interior angles =
Complete the following table:
Polygon name Number of sides Sum of measures of interior angles
triangle
octagon
decagon
A regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each angle in a regular pentagon?
What is the measure of each angle in a regular octagon?
Can you generalize your answer from 6 and 7? What is the measure of each angle in a regular gon?
Can you use the polygon angle sum theorem on a convex polygon? Why or why not? Use the convex quadrilateral to explain your answer.
If we know the sum of the angles in a polygon is , how many sides does the polygon have? Show the work leading to your answer.
Review Answers
One possible answer:
Number of sides , sum of interior angles
Number of sides , sum of interior angles
Polygon name Number of sides Sum of measures of interior angles
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
decagon
dodecagon
gon
Since the sum of the angles is , each angle measures
Answers will vary. One possibility is no, we cannot use the polygon angle sum theorem because is an acute angle that does not open inside the polygon. Alternatively, if we allow for angles between and , then we can use the angle sum theorem, but so far we have not seen angles measuring more than
Solve the equation:
Exterior Angles
Learning Objectives
Identify the exterior angles of convex polygons.
Find the sums of exterior angles in convex polygons.
Introduction
This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior angles in a polygon that will help you solve problems about regular polygons.
Exterior Angles in Convex Polygons
Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the outside of a polygon. An exterior angle is formed by extending a side of the polygon.
As you can tell, there are two possible exterior angles for any given vertex on a polygon. In the figure above we only showed one set of exterior angles; the other set would be formed by extending each side in the opposite (clockwise) direction. However, it doesn’t matter which exterior angles you use because on each vertex their measurement will be the same. Let’s look closely at one vertex, and draw both of the exterior angles that are possible.
As you can see, the two exterior angles at the same vertex are vertical angles. Since vertical angles are congruent, the two exterior angles possible around a single vertex are congruent.
Additionally, because the exterior angle will be a linear pair w
ith its adjacent interior angle, it will always be supplementary to that interior angle. As a reminder, supplementary angles have a sum of .
Example 1
What is the measure of the exterior angle in the diagram below?
The interior angle is labeled as . Since you need to find the exterior angle, notice that the interior angle and the exterior angle form a linear pair. Therefore the two angles are supplementary—they sum to . So, to find the measure of the exterior angle, subtract from .
The measure of is .
Summing Exterior Angles in Convex Polygons
CK-12 Geometry Page 21
