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Triangle Proof Worksheets

Proofs are used in geometry to provide evidence that a math statement is to be seen as true. They can be presented in several different fashions, but the overall goal is providing the reader a set of logical facts that leads them to accept something as true. When it comes to using proofs with triangles, we are often trying to have our readers conclude that two triangles are congruent. There are five common postulates (SSS, SAS, ASA, AAS, and HL) that we use to do this. In most cases the end goal is to prove that two angles or sides share the same measure. These worksheets and lessons help students to learn how to start writing and creating proofs of triangles of their very own. This takes some practice and can be intimidating at first, but it gets easy with practice.

Aligned Standard: High School Geometry - HSG-CO.C.10

  • Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Bisectors appear for the first time in this section.

  • Homework 1 - Use that that given so that you can prove that line PR bisects ∠QPS & ∠QRS. You can use the theorem ASA.
  • Homework 2 - When two congruent angles are bisected, it leaves you with a lot about the system. Use it to your advantage.
  • Homework 3 - It is interesting how angles from different sides of the track are congruent.

Practice Worksheets

Remind students that drawings are not drawn to scale. This is important to keep in mind.

  • Practice 1 - Line VY is a mid segment of the ▲ UWX. If WX= 6, Prove VY=3. This throws them at first, just remind them it is not to scale.
  • Practice 2 - Midpoints are found at the halfway mark. We learn the difference between midpoints and midsegments.
  • Practice 3 - UW is a mid segment of ▲ VXY. If UW = 5, prove XY= 10

Math Skill Quizzes

These are some of the most open ended questions that you will see in geometry.

  • Quiz 1 - WXYZ is parallelogram. What does that tell us about WX and YZ?
  • Quiz 2 - What is the condition of a bisector divides a segment into equal halves?
  • Quiz 3 - What does a rhombus tell you about the shape found on the other side?

What are Triangle Proofs?

Goofball Triangle

A proof in geometry is a statement or series of them that proves a math concept to be true. They consist of a group of reasons and statements. Proofs normally are setup in three different ways. The two-column proof is the most popular method. In this technique you draw up two columns with a large “T” shape. In the first column (left) math statements are written. In the second column (right) corresponding reasons are listed. Proofs can also be written in straight paragraph form. This is where math statements are written and followed immediately by reasons. The last method that you will see more at the college level is a flow chart. This is a much more visual method in that it uses boxes and arrows. Statements are written inside the boxes and the reasons are written beneath the boxes.

When it comes to using proofs with triangles there are number of different reasons you may want to write them. We commonly use proofs to prove that two triangles are congruent. Since the sum of the angles with a triangle are 180 degrees, we often use this fact to find missing angles within triangles or those congruent to. Besides angles, we often use the Pythagorean Theorem, among other trigonometric theorems to find missing sides between congruent triangles.

The Anatomy of Triangle Proofs

A triangle is a three-sided polygon which can be classified by its angles and sides. When classifying we are based on their sides, you should see if any sides are of the same lengths. If none of the sides are equal, then it is known as scalene. If any two lengths are the same, then it is known as isosceles. If all three sides are of the same length, then it is known as the equilateral.

When classifying the triangle based on angles, you look at the size of angles. If the triangle has a 90° , then it is known as a right. If all the angles of a triangle measure less than 90° then it is known as an acute. And if the measure of one of the angles is greater than 90° , then it is known as an obtuse.

If you were asked to prove that sum of the interior angles of a triangle is 180 degrees, you would need to be familiar with this property of a triangle. Below we have discussed the proof for this property.

Consider a triangle ABC: Using parallel postulate, there is one line parallel to side AC passing through B. ∠DBA is congruent to ∠A as they are alternate interior angles, and these angles are congruent when lines are parallel. Therefore, ∠DBA is equal to ∠A. Similarly, ∠EBC is congruent to ∠C as they are also alternate interior angles and so ∠EBC is equal to ∠C. ∠DBA + ∠ABC + ∠EBC = 180.

Tips for Working with Triangle Proofs

You always what to make sure that you completely consider all the information that you are provided with prior to starting anything. This is often called the given, make sure to rewrite it. This will often be the first step of these types of problems. It may require you to rearrange and restate this material evidence slightly differently than it has been presented to you. In most cases, you will start by proving some form of congruency which can be inferred from the given. If congruency is not blatantly provided for you in the given, so what you can relate to the given to make it so. This can be done through a number of different theorems that relate to triangle geometry such as ASA, AAS, HL, SAS, or SSS. Once congruency is proven, you can infer the measures of related parts by just referring to basis of theorem itself. The foundation is based on the fact that corresponding parts of congruent triangles are themselves congruent. In general, you will find that following this approach will help you provide solid proofs four out five times.

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