angle proofs homework 8

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High school geometry

Course: high school geometry   >   unit 3.

• Geometry proof problem: midpoint
• Geometry proof problem: congruent segments
• Geometry proof problem: squared circle

Line and angle proofs

• W U ― ‍   perpendicular to T V ↔ ‍   such that U ‍   is on T V ↔ ‍  
• V Y ― ‍   perpendicular to W X ↔ ‍   such that Y ‍   is on W X ↔ ‍  
• (Choice A)   When a transversal crosses parallel lines, alternate interior angles are congruent. A When a transversal crosses parallel lines, alternate interior angles are congruent.
• (Choice B)   When a transversal crosses parallel lines, same-side interior angles are congruent. B When a transversal crosses parallel lines, same-side interior angles are congruent.
• (Choice C)   Angles that form a linear pair are supplementary. C Angles that form a linear pair are supplementary.
• (Choice D)   Vertical angles are congruent. D Vertical angles are congruent.

Unknown Angle Proofs — Writing Proofs

Related Topics: Lesson Plans and Worksheets for Geometry Lesson Plans and Worksheets for all Grades More Lessons for Geometry Common Core For Geometry

New York State Common Core Math Geometry, Module 1, Lesson 9

Worksheets for Geometry

Student Outcomes

• Students write unknown angle proofs, which does not require any new geometric facts. Rather, writing proofs requires students to string together facts they already know to reveal more information.

Opening Exercise

One of the main goals in studying geometry is to develop your ability to reason critically, to draw valid conclusions based upon observations and proven facts. Master detectives do this sort of thing all the time. Take a look as Sherlock Holmes uses seemingly insignificant observations to draw amazing conclusions.

Could you follow Sherlock Holmes’s reasoning as he described his thought process?

In geometry, we follow a similar deductive thought process (much like Holmes uses) to prove geometric claims. Let’s revisit an old friend—solving for unknown angles. Remember this one?

You needed to figure out the measure of 𝑎 and used the “fact” that an exterior angle of a triangle equals the sum of the measures of the opposite interior angles. The measure of ∠𝑎 must, therefore, be 36°.

Suppose that we rearrange the diagram just a little bit. Instead of using numbers, we use variables to represent angle measures.

Suppose further that we already know that the angles of a triangle sum to 180°. Given the labeled diagram to the right, can we prove that 𝑥 + 𝑦 = 𝑧 (or, in other words, that the exterior angle of a triangle equals the sum of the measures of the opposite interior angles)?

PROOF: Label ∠𝑤, as shown in the diagram. 𝑚∠𝑥 + 𝑚∠𝑦 + 𝑚∠𝑤 = 180° The sum of the angle measures in a triangle is 180°. 𝑚∠𝑤 + 𝑚∠𝑧 = 180° Linear pairs form supplementary angles. 𝑚∠𝑥 + 𝑚∠𝑦 + 𝑚∠𝑤 = 𝑚∠𝑤 + 𝑚∠𝑧 Substitution property of equality ∴ 𝑚∠𝑥 + 𝑚∠𝑦 = 𝑚∠𝑧

Notice that each step in the proof was justified by a previously known or demonstrated fact. We end up with a newly proven fact (that an exterior angle of any triangle is the sum of the measures of the opposite interior angles of the triangle). This ability to identify the steps used to reach a conclusion based on known facts is deductive reasoning (i.e., the same type of reasoning that Sherlock Holmes used to accurately describe the doctor’s attacker in the video clip).

Exercises 1–6

• You know that angles on a line sum to 180°. Prove that vertical angles are equal in measure. Make a plan:
• What do you know about ∠𝑤 and ∠𝑥? ∠𝑦 and ∠𝑥?
• What conclusion can you draw based on both pieces of knowledge?
• Write out your proof:
• Given the diagram to the right, prove that 𝑚∠𝑤 + 𝑚∠𝑥 + 𝑚∠𝑧 = 180°. (Make a plan first. What do you know about ∠𝑥, ∠𝑦, and ∠𝑧?) Given the diagram to the right, prove that 𝑚∠𝑤 = 𝑚∠𝑦 + 𝑚∠𝑥
• In the diagram to the right, prove that 𝑚∠𝑦 + 𝑚∠𝑧 = 𝑚∠𝑤 + 𝑚∠𝑥. (You need to write a label in the diagram that is not labeled yet for this proof.)
• In the figure to the right, 𝐴𝐵 || 𝐶𝐷 and 𝐵𝐶 || 𝐷𝐸. Prove that 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐶𝐷𝐸.
• In the figure to the right, prove that the sum of the angles marked by arrows is 900°. (You need to write several labels in the diagram for this proof.)
• In the figure to the right, prove that 𝐷𝐶 ⊥ 𝐸𝐹. Draw in label 𝑍.

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Angle Properties, Postulates, and Theorems

In order to study geometry in a logical way, it will be important to understand key mathematical properties and to know how to apply useful postulates and theorems. A postulate is a proposition that has not been proven true, but is considered to be true on the basis for mathematical reasoning. Theorems , on the other hand, are statements that have been proven to be true with the use of other theorems or statements. While some postulates and theorems have been introduced in the previous sections, others are new to our study of geometry. We will apply these properties, postulates, and theorems to help drive our mathematical proofs in a very logical, reason-based way.

Before we begin, we must introduce the concept of congruency. Angles are congruent if their measures, in degrees, are equal. Note : “congruent” does not mean “equal.” While they seem quite similar, congruent angles do not have to point in the same direction. The only way to get equal angles is by piling two angles of equal measure on top of each other.

We will utilize the following properties to help us reason through several geometric proofs.

Reflexive Property

A quantity is equal to itself.

Symmetric Property

If A = B , then B = A .

Transitive Property

If A = B and B = C , then A = C .

Addition Property of Equality

If A = B , then A + C = B + C .

Angle Postulates

Angle addition postulate.

If a point lies on the interior of an angle, that angle is the sum of two smaller angles with legs that go through the given point.

Consider the figure below in which point T lies on the interior of ?QRS . By this postulate, we have that ?QRS = ?QRT + ?TRS . We have actually applied this postulate when we practiced finding the complements and supplements of angles in the previous section.

Corresponding Angles Postulate

If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.

Converse also true : If a transversal intersects two lines and the corresponding angles are congruent, then the lines are parallel.

The figure above yields four pairs of corresponding angles.

Parallel Postulate

Given a line and a point not on that line, there exists a unique line through the point parallel to the given line.

The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry .

There are an infinite number of lines that pass through point E , but only the red line runs parallel to line CD . Any other line through E will eventually intersect line CD .

Angle Theorems

Alternate exterior angles theorem.

If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.

Converse also true : If a transversal intersects two lines and the alternate exterior angles are congruent, then the lines are parallel.

The alternate exterior angles have the same degree measures because the lines are parallel to each other.

Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then the alternate interior angles are congruent.

Converse also true : If a transversal intersects two lines and the alternate interior angles are congruent, then the lines are parallel.

The alternate interior angles have the same degree measures because the lines are parallel to each other.

Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Right Angles Theorem

All right angles are congruent.

Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

Converse also true : If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

The sum of the degree measures of the same-side interior angles is 180°.

Vertical Angles Theorem

If two angles are vertical angles, then they have equal measures.

The vertical angles have equal degree measures. There are two pairs of vertical angles.

(1) Given: m?DGH = 131

Find: m?GHK

First, we must rely on the information we are given to begin our proof. In this exercise, we note that the measure of ?DGH is 131° .

From the illustration provided, we also see that lines DJ and EK are parallel to each other. Therefore, we can utilize some of the angle theorems above in order to find the measure of ?GHK .

We realize that there exists a relationship between ?DGH and ?EHI : they are corresponding angles. Thus, we can utilize the Corresponding Angles Postulate to determine that ?DGH??EHI .

Directly opposite from ?EHI is ?GHK . Since they are vertical angles, we can use the Vertical Angles Theorem , to see that ?EHI??GHK .

Now, by transitivity , we have that ?DGH??GHK .

Congruent angles have equal degree measures, so the measure of ?DGH is equal to the measure of ?GHK .

Finally, we use substitution to conclude that the measure of ?GHK is 131° . This argument is organized in two-column proof form below.

(2) Given: m?1 = m?3

Prove: m?PTR = m?STQ

We begin our proof with the fact that the measures of ?1 and ?3 are equal.

In our second step, we use the Reflexive Property to show that ?2 is equal to itself.

Though trivial, the previous step was necessary because it set us up to use the Addition Property of Equality by showing that adding the measure of ?2 to two equal angles preserves equality.

Then, by the Angle Addition Postulate we see that ?PTR is the sum of ?1 and ?2 , whereas ?STQ is the sum of ?3 and ?2 .

Ultimately, through substitution , it is clear that the measures of ?PTR and ?STQ are equal. The two-column proof for this exercise is shown below.

(3) Given: m?DCJ = 71 , m?GFJ = 46

Prove: m?AJH = 117

We are given the measure of ?DCJ and ?GFJ to begin the exercise. Also, notice that the three lines that run horizontally in the illustration are parallel to each other. The diagram also shows us that the final steps of our proof may require us to add up the two angles that compose ?AJH .

We find that there exists a relationship between ?DCJ and ?AJI : they are alternate interior angles. Thus, we can use the Alternate Interior Angles Theorem to claim that they are congruent to each other.

By the definition of congruence , their angles have the same measures, so they are equal.

Now, we substitute the measure of ?DCJ with 71 since we were given that quantity. This tells us that ?AJI is also 71° .

Since ?GFJ and ?HJI are also alternate interior angles, we claim congruence between them by the Alternate Interior Angles Theorem .

The definition of congruent angles once again proves that the angles have equal measures. Since we knew the measure of ?GFJ , we just substitute to show that 46 is the degree measure of ?HJI .

As predicted above, we can use the Angle Addition Postulate to get the sum of ?AJI and ?HJI since they compose ?AJH . Ultimately, we see that the sum of these two angles gives us 117° . The two-column proof for this exercise is shown below.

(4) Given: m?1 = 4x + 9 , m?2 = 7(x + 4)

In this exercise, we are not given specific degree measures for the angles shown. Rather, we must use some algebra to help us determine the measure of ?3 . As always, we begin with the information given in the problem. In this case, we are given equations for the measures of ?1 and ?2 . Also, we note that there exists two pairs of parallel lines in the diagram.

By the Same-Side Interior Angles Theorem , we know that that sum of ?1 and ?2 is 180 because they are supplementary.

After substituting these angles by the measures given to us and simplifying, we have 11x + 37 = 180 . In order to solve for x , we first subtract both sides of the equation by 37 , and then divide both sides by 11 .

Once we have determined that the value of x is 13 , we plug it back in to the equation for the measure of ?2 with the intention of eventually using the Corresponding Angles Postulate . Plugging 13 in for x gives us a measure of 119 for ?2 .

Finally, we conclude that ?3 must have this degree measure as well since ?2 and ?3 are congruent . The two-column proof that shows this argument is shown below.

• Finding values in an expansion of the ascending powers of x
• Evaluation of a decimal number raised to a power
• Solving of Trigonometric Ratio using a Pythagorean Identity
• Solving of composite function

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Unit 2 Logic And Proofs Homework 8 Segment Proofs

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