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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 180∘.

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 𝑂, defined as 𝑂𝐴, οƒͺ𝑂𝐡, 𝑂𝐢, and 𝑂𝐷.

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

Since the angle measures on a straight line sum to 180∘, we have that π‘šβˆ π΄π‘‚π΅+π‘šβˆ π΅π‘‚πΆ+π‘šβˆ πΆπ‘‚πΈ=180.∘

We can also write that π‘šβˆ π΄π‘‚π·+π‘šβˆ π·π‘‚πΈ=180∘ since these are also angles that lie on a straight line.

Therefore, the sum of all the angles about point 𝑂 can be given as π‘šβˆ π΄π‘‚π΅+π‘šβˆ π΅π‘‚πΆ+π‘šβˆ πΆπ‘‚πΈ+π‘šβˆ π΄π‘‚π·+π‘šβˆ π·π‘‚πΈ=180+180=360.∘∘∘

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 360∘.

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 180∘.

Example 1: Completing the Proof for Vertically Opposite Angles

Two straight lines, ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷, intersect at point 𝐸.

  1. Fill in the blank: If the angles ∠𝐴𝐸𝐷 and ∠𝐴𝐸𝐢 are adjacent angles where οƒͺ𝐸𝐢βˆͺ𝐸𝐷=𝐢𝐷, then π‘šβˆ π΄πΈπΆ+π‘šβˆ π΄πΈπ·=.
  2. Fill in the blank: If the angles ∠𝐴𝐸𝐢 and ∠𝐢𝐸𝐡 are adjacent angles where 𝐸𝐴βˆͺοƒͺ𝐸𝐡=𝐴𝐡, then π‘šβˆ π΄πΈπΆ+π‘šβˆ πΆπΈπ΅=.
  3. True or False: We deduce from the two parts above that π‘šβˆ π΄πΈπ·=π‘šβˆ πΆπΈπ΅.

Answer

We can begin by drawing ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷, which intersect at a point 𝐸.

Part 1

We then need to consider ∠𝐴𝐸𝐢 and ∠𝐴𝐸𝐷.

We recall that the angle measures on a straight line sum to 180∘, and since these two angles lie on ⃖⃗𝐢𝐷, then their measures must sum to 180∘.

Therefore, we can fill in the first blank: If the angles ∠𝐴𝐸𝐷 and ∠𝐴𝐸𝐢 are adjacent angles where οƒͺ𝐸𝐢βˆͺ𝐸𝐷=𝐢𝐷, then π‘šβˆ π΄πΈπΆ+π‘šβˆ π΄πΈπ·=180∘.

Part 2

Next, we consider ∠𝐴𝐸𝐢 and ∠𝐢𝐸𝐡.

Once again, since the angle measures on a straight line sum to 180∘, then we know that π‘šβˆ π΄πΈπΆ+π‘šβˆ πΆπΈπ΅=180∘. Hence, we can complete the second missing blank.

If the angles ∠𝐴𝐸𝐢 and ∠𝐢𝐸𝐡 are adjacent angles where 𝐸𝐴βˆͺοƒͺ𝐸𝐡=𝐴𝐡, then π‘šβˆ π΄πΈπΆ+π‘šβˆ πΆπΈπ΅=180∘.

Part 3

We now consider ∠𝐴𝐸𝐷 and ∠𝐢𝐸𝐡.

It may be useful if we return to the angles in the first part of this question and label π‘šβˆ π΄πΈπ· as π‘₯∘ and π‘šβˆ π΄πΈπΆ as π‘¦βˆ˜.

We know that π‘₯+𝑦=180.∘∘∘

Hence, when we consider the second pair of angles, ∠𝐴𝐸𝐢 and ∠𝐢𝐸𝐡, π‘šβˆ πΆπΈπ΅ must also be π‘₯∘.

We can then observe that π‘šβˆ π΄πΈπ·=π‘šβˆ πΆπΈπ΅. 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 π‘šβˆ π΄πΈπΆ=π‘šβˆ π·πΈπ΅ since these would both be equal to π‘¦βˆ˜.

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.

Answer

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

Given that ⃖⃗𝐸𝐹 is perpendicular to ⃖⃗𝐴𝐡, we can write that π‘šβˆ πΈπ‘ƒπ΅=90.∘

Given that ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷 are parallel, we know that corresponding angles are congruent. βˆ πΈπ‘„π· is corresponding to βˆ πΈπ‘ƒπ΅; hence, π‘šβˆ πΈπ‘„π·=π‘šβˆ πΈπ‘ƒπ΅=90.∘

Thus, ⃖⃗𝐸𝐹 is also perpendicular to ⃖⃗𝐢𝐷.

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 ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷 are parallel.

Answer

In the figure below, we can observe that π‘šβˆ π·π‘πΉ=75∘. We recall that, when two straight lines intersect, the vertically opposite angles are equal in measure. As ⃖⃗𝐷𝐢 intersects ⃖⃗𝐹𝐸, we have π‘šβˆ πΈπ‘πΆ=π‘šβˆ π·π‘πΉ=75.∘

We then observe that we have another pair of congruent angle measures: π‘šβˆ πΈπ‘πΆ=π‘šβˆ πΈπ‘€π΄=75.∘

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 ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷 are parallel is true.

Note that we could have alternatively demonstrated that βˆ π΅π‘€πΉ and βˆ πΈπ‘€π΄ are vertically opposite and equal in measure and that π‘šβˆ π΅π‘€πΉ is equal to π‘šβˆ π·π‘πΉ.

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 ∠𝐡𝐢𝐷.

Answer

In the figure, we can observe that there are two pairs of congruent lines marked as follows: 𝐴𝐡=𝐴𝐷,𝐢𝐡=𝐢𝐷.

If we consider △𝐴𝐡𝐢 and △𝐴𝐷𝐢, we can also note that 𝐴𝐢=𝐴𝐢.(sincethisisacommonsidetoboth)

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

We are given that π‘šβˆ π΄πΆπ·=29∘. The corresponding angle, ∠𝐴𝐢𝐡, has equal measure. That is π‘šβˆ π΄πΆπ΅=π‘šβˆ π΄πΆπ·=29.∘

In this question, we need to calculate π‘šβˆ π΅πΆπ·. ∠𝐡𝐢𝐷 is comprised of ∠𝐴𝐢𝐡 and ∠𝐴𝐢𝐷. Hence, we have π‘šβˆ π΅πΆπ·=π‘šβˆ π΄πΆπ΅+π‘šβˆ π΄πΆπ·=29+29=58.∘∘∘

Therefore, we have used the properties of congruent triangles to calculate that π‘šβˆ π΅πΆπ· is 58∘.

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, π‘šβˆ π΅π΄πΆ=π‘šβˆ π΅π·πΆ=50∘ and π‘šβˆ πΆπ΅π·=π‘šβˆ π΄πΆπ΅=40∘.

  1. Is the triangle 𝐴𝐡𝐢 congruent with the triangle 𝐷𝐢𝐡?
  2. Is the diagonal 𝐴𝐢 equal to the diagonal 𝐡𝐷?

Answer

Part 1

In order to consider the two triangles △𝐴𝐡𝐢 and △𝐷𝐢𝐡, 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 180∘.

Hence, in △𝐷𝐢𝐡, given that π‘šβˆ πΆπ΅π·=40∘ and π‘šβˆ π΅π·πΆ=50∘, we have π‘šβˆ π·πΆπ΅+π‘šβˆ πΆπ΅π·+π‘šβˆ π΅π·πΆ=180π‘šβˆ π·πΆπ΅+40+50=180π‘šβˆ π·πΆπ΅+90=180π‘šβˆ π·πΆπ΅=180βˆ’90=90.∘∘∘∘∘∘∘∘∘

Similarly, for △𝐴𝐡𝐢, given that π‘šβˆ π΄πΆπ΅=40∘ and π‘šβˆ π΅π΄πΆ=50∘, we have π‘šβˆ π΄π΅πΆ+π‘šβˆ π΄πΆπ΅+π‘šβˆ π΅π΄πΆ=180π‘šβˆ π΄π΅πΆ+40+50=180π‘šβˆ π΄π΅πΆ+90=180π‘šβˆ π΄π΅πΆ=180βˆ’90=90.∘∘∘∘∘∘∘∘∘

We can then recognize that there is a common side in both triangles: 𝐢𝐡=𝐡𝐢.

We now have enough information to demonstrate that △𝐴𝐡𝐢≅△𝐷𝐢𝐡 using the ASA criterion since π‘šβˆ π·πΆπ΅=π‘šβˆ π΄π΅πΆ,𝐢𝐡=𝐡𝐢,π‘šβˆ π·π΅πΆ=π‘šβˆ π΄πΆπ΅.(angle)(includedside)(angle)

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

Part 2

Consider the diagonals of the figure, 𝐴𝐢 and 𝐡𝐷.

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

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

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

We can consider △𝐴𝐡𝐷 and △𝐷𝐢𝐴, and, as we determined that △𝐴𝐡𝐢≅△𝐷𝐢𝐡, we have that 𝐴𝐡=𝐷𝐢,𝐴𝐢=𝐷𝐡,𝐴𝐷.andisacommonside

Hence, we can say that △𝐴𝐡𝐷≅△𝐷𝐢𝐴.

In 𝐴𝐡𝐢𝐷, there are two pairs of opposite sides that are congruent (𝐴𝐷=𝐢𝐡 and 𝐢𝐷=𝐴𝐡). So, 𝐴𝐡𝐢𝐷 must be a parallelogram. However, we can further note that, as π‘šβˆ π΄π΅πΆ=π‘šβˆ π·πΆπ΅=90∘, then 𝐴𝐡𝐢𝐷 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 360∘.
    • If two straight lines intersect, then the vertically opposite angles are equal in measure.
    • In a rectangle, the diagonals are equal in length.

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