# Lesson Explainer: Deductive Proof for Geometric Properties Mathematics

In this explainer, we will learn how to prove certain geometric properties using deductive proof.

Let’s begin by recapping an important geometric property.

### Recap: Angles on a Straight Line

The angle measures on a straight line sum to .

We will see how we can use this geometric property to prove other geometric properties.

Let’s consider the sum of the measures of the angles around a point. For example, we can create the rays from a point , defined as , , , and .

We want to determine the sum of the angle measures around . We can do this by constructing the line , which passes through .

Since the angle measures on a straight line sum to , we have that

We can also write that since these are also angles that lie on a straight line.

Therefore, the sum of all the angles about point can be given as

This property can be defined as below.

### Theorem: Sum of the Measures of Angles around a Point

The sum of all the measures of angles around a point is .

Let’s now recall what vertically opposite angles are.

### Definition: Vertically Opposite Angles

Vertically opposite angles are the angles created when two straight lines intersect.

Vertically opposite angles have the following important property.

### Property: Vertically Opposite Angles

If two straight lines intersect, the vertically opposite angles, sometimes referred to as opposite angles, are equal in measure.

We should already be familiar with finding angle measures by using this property. In the following example, we will see how we can prove this by using the property that the angle measures on a straight line sum to .

### Example 1: Completing the Proof for Vertically Opposite Angles

Two straight lines, and , intersect at point .

1. Fill in the blank: If the angles and are adjacent angles where , then .
2. Fill in the blank: If the angles and are adjacent angles where , then .
3. True or False: We deduce from the two parts above that .

We can begin by drawing and , which intersect at a point .

Part 1

We then need to consider and .

We recall that the angle measures on a straight line sum to , and since these two angles lie on , then their measures must sum to .

Therefore, we can fill in the first blank: If the angles and are adjacent angles where , then .

Part 2

Next, we consider and .

Once again, since the angle measures on a straight line sum to , then we know that . Hence, we can complete the second missing blank.

If the angles and are adjacent angles where , then .

Part 3

We now consider and .

It may be useful if we return to the angles in the first part of this question and label as and as .

We know that

Hence, when we consider the second pair of angles, and , must also be .

We can then observe that . Hence, the statement is true.

In the previous example, we proved that a pair of vertically opposite angles are equal, and, in fact, in this example, we could also say that since these would both be equal to .

In the following example, we will see how we can apply the property of vertically opposite angles along with other geometric properties.

First, we recap some important facts about congruent and supplementary angle measures in parallel lines.

### Property: Angles in a Set of Parallel Lines

When a pair of parallel lines is intersected by another line known as a transversal, it creates pairs of congruent or supplementary angle measures.

We will use these angle properties in the next example.

### Example 2: Proving a Geometric Statement Using Parallel Lines and Traversals

True or False: A straight line that is perpendicular to one of two parallel lines is also perpendicular to the other.

We can begin by drawing two parallel lines, labeled and , with a straight line that is perpendicular to . We can label the points where intersects and as and respectively.

Given that is perpendicular to , we can write that

Given that and are parallel, we know that corresponding angles are congruent. is corresponding to ; hence,

Thus, is also perpendicular to .

We can give the answer that the statement, a straight line that is perpendicular to one of two parallel lines is also perpendicular to the other, is true.

Let’s now see another example involving parallel lines.

### Example 3: Completing a Geometric Proof Using Vertically Opposite Angles

True or False: In the given figure, the two straight lines and are parallel.

In the figure below, we can observe that . We recall that, when two straight lines intersect, the vertically opposite angles are equal in measure. As intersects , we have

We then observe that we have another pair of congruent angle measures:

Hence, these two angles are corresponding angles. We recall that, if the corresponding angles that a transversal makes with a pair of lines are congruent, then the pair of lines is parallel. Therefore, the statement that and are parallel is true.

Note that we could have alternatively demonstrated that and are vertically opposite and equal in measure and that is equal to .

These two corresponding angles would also demonstrate that the statement is true.

We can also recall that there are a number of geometric properties that we have learned about congruent triangles. Let’s recap what congruent triangles are and how we can prove that two triangles are congruent.

### Definition: Congruent Triangles and Congruence Criteria

Two triangles are congruent if their corresponding sides are congruent and their corresponding angle measures are congruent. The congruence criteria allow us to more easily prove if triangles are congruent. They are as follows:

• Side-angle-side (SAS): Two triangles are congruent if they have two sides that are congruent and the included angle is congruent.
• Angle-side-angle (ASA): Two triangles are congruent if they have two angles that are congruent and the included side is congruent.
• Side-side-side (SSS): Two triangles are congruent if they have all three sides congruent.
• Right angle-hypotenuse-side (RHS): Two triangles are congruent if they both have a right angle and the hypotenuse and one other side are congruent.

Just like many of the other geometric facts, being able to recall and use the criteria for congruent triangles can be an excellent tool in enabling us to prove many different geometric properties. In fact, the application of congruent triangles is used in many proofs about the angles and side lengths in polygons that we may already know.

We will see an example of this below.

### Example 4: Using Congruent Triangles to Prove Geometric Statements

In the given figure, use the properties of congruent triangles to find the measure of .

In the figure, we can observe that there are two pairs of congruent lines marked as follows:

If we consider and , we can also note that

Thus, in and we have three congruent pairs of sides. This means that is congruent to by the SSS congruence criterion. Therefore, all corresponding sides and angles are congruent.

We are given that . The corresponding angle, , has equal measure. That is

In this question, we need to calculate . is comprised of and . Hence, we have

Therefore, we have used the properties of congruent triangles to calculate that is .

In the previous example, we found an unknown angle measure. However, this question also demonstrates the proof of a geometric property: the longer diagonal of a kite bisects the angles at the vertices on this diagonal.

We will continue with the use of congruent triangles to demonstrate a geometric property in the final example.

### Example 5: Proving a Geometric Property Using Congruent Triangles

In the given figure, and .

1. Is the triangle congruent with the triangle ?
2. Is the diagonal equal to the diagonal ?

Part 1

In order to consider the two triangles and , it can be helpful to draw them separately and label the vertices and the given measures.

We can observe that the third angle measure in each triangle can be calculated by recalling that the sum of the interior angle measures in a triangle equals .

Hence, in , given that and , we have

Similarly, for , given that and , we have

We can then recognize that there is a common side in both triangles:

We now have enough information to demonstrate that using the ASA criterion since

Therefore, we can give the following answer: yes, triangle is congruent with triangle .

Part 2

Consider the diagonals of the figure, and .

As we have proved that , we know that the corresponding sides in the triangle are congruent. In and , these two line segments forming the diagonals are corresponding.

Hence, and are equal in length. Therefore, we can answer the second part of this question as follows: yes, the diagonal is equal to the diagonal .

Let’s now consider the quadrilateral in the previous question more closely. We will seek to demonstrate that is a rectangle.

We can consider and , and, as we determined that , we have that

Hence, we can say that .

In , there are two pairs of opposite sides that are congruent ( and ). So, must be a parallelogram. However, we can further note that, as , then is a rectangle.

The working in this previous example demonstrates the property that, in a rectangle, the diagonals are equal in length.

We now summarize the key points.

### Key Points

• We can use geometric properties to prove further geometric properties as part of a deductive proof.
• We used geometric properties to prove the following:
• The sum of all the angle measures around a point is .
• If two straight lines intersect, then the vertically opposite angles are equal in measure.
• In a rectangle, the diagonals are equal in length.