Geometry theorems

Here you will learn about geometry theorems, including the angle sum theorem, vertical angles theorem, alternate interior angles theorem, exterior angle theorem and the Pythagorean theorem.

Students will first learn about geometry theorems as part of geometry in 7 th and 8 th grade and continue to learn about them in high school.

What are geometry theorems?

Geometry theorems are statements, related to geometry, that have been proven to be true through logic and are accepted to be mathematical principles.

Note: This page will overview geometry theorems and how to use them, but does not explore the proofs of these theorems.

The angle sum theorem (triangle sum theorem) says that the interior angles of any triangle will always have a sum of 180^{\circ}.

For example,

See also: Angles of a triangle

The vertical angles theorem (opposite angles) says when two lines intersect once they form two pairs of congruent angles.

For example,

Step-by-step guide: Vertical angles theorem

The alternate interior angles theorem says when a transversal passes through two parallel lines, angles on opposite sides of the transversal, but inside the parallel lines are congruent (equal in measure).

For example,

See also: Alternate interior angles theorem

The exterior angle theorem says the angle formed by extending a side of the triangle is equal to the sum of the other two non-adjacent angles in the triangle.

For example,

The Pythagorean theorem says the square of the hypotenuse of a right triangle (the longest side) is equal to the sum of the squares of the other two sides.

For example,

See also: Pythagorean theorem

What are geometry theorems?

Common Core State Standards

How does this relate to 7 th grade math and 8 th grade math?

• Grade 7 – Geometry (7.G.B.5)
  Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

• Grade 8 – Geometry (8.G.A.5)
  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

• Grade 8 – Geometry (8.G.B.6)
  Explain a proof of the Pythagorean Theorem and its converse.

• Grade 8 – Geometry (8.G.B.7)
  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

How to use geometry theorems to solve problems

In order to use geometry theorems to solve problems:

1. Decide which theorem is needed to solve.
2. Use the theorem to solve.

Geometry theorem examples

Example 1: missing angle in a triangle

Find the measure of the unknown angle labeled b in the following triangle:

1. Decide which theorem is needed to solve.

The question is asking about the total measurement of the interior angles of a triangle. This is the angles sum theorem.

2Use the theorem to solve.

The angles sum theorem says that all interior angles of a triangle sum to be 180^{\circ}.

Add up the angles that are given within the triangle.The angles 90^{\circ} and 22^{\circ} are given. Add these together:

Subtract 112^{\circ} from 180^{\circ}\text{:}

Example 2: missing angles in intersecting lines

Find the values of angles x and y.

Example 3: missing angle in a parallel lines cut by a transversal

Calculate the size of the missing angle \theta. Justify your answer.

Use the theorem to solve.

Here you can label the alternate angle on the diagram as 48^{\circ}.

Use basic angle facts to calculate the other missing angle.

Here as \theta is on a straight line with 48^{\circ},

\theta=180-48=132^{\circ}

Example 4: missing exterior angle in a triangle

Calculate the value of x.

Example 5: missing exterior angle in a triangle

Calculate the value of x.

Example 6: missing length of the hypotenuse c

Find the third side, x, and answer to the nearest hundredth.

Use the theorem to solve.

Label the hypotenuse (the longest side) with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

Write down the formula and apply the numbers.

\begin{aligned}& a^2+b^2=c^2 \\\\ & 3^2+9^2=x^2 \\\\ & 9+81=x^2 \\\\ & 90=x^2 \\\\ & \sqrt{90}=x \end{aligned}

An alternative method of rearranging the formula and to put one calculation into a calculator will also work.

\begin{aligned}& a^2+b^2=c^2 \\\\ & c^2=a^2+b^2 \\\\ & c=\sqrt{a^2+b^2} \\\\ & x=\sqrt{3^2+9^2} \end{aligned}

Make sure you give your final answer in the correct form by calculating the square root value; including units where appropriate.

x=\sqrt{90}=9.48683298 \ldots

The final answer to the nearest hundredth is:

x=9.49 \mathrm{~cm}

Teaching tips for geometry theorems

• Introduce all geometric theorems with whole numbers, such as whole number sides of a triangle or angle measurements, so that students can focus on the qualities of the theorem and not the calculations.

• Once students understand all the theorems, challenge them to prove angle congruence using a mixture of the theorems.

Easy mistakes to make

• Incorrectly labeling angles
  Write neatly, so that you can keep track of each angle measurement and avoid mistakes.

• Assuming corresponding sides or angles are equal, because they “look” equal
  It is important to remember that the images are not to scale, so just because an angle “looks” like a certain measurement, does not make it true. Always use only the information given to solve, instead of measuring with a ruler endpoint to endpoint or using a protractor.

• Mixing up the terminology for the theorems
  This is particularly common with theorems that have similar names. There are different rules for the same-side interior angles theorem, than the alternate interior angles theorem. Always pay close attention to the theorem name and the angles given in a problem.

Related angles lessons

• Measuring angles
• Acute angle
• Obtuse angle
• Right angle
• Adjacent angle
• Supplementary angles
• Complementary angles
• Straight angle
• Angles point
• Pentagon angle

The next lessons are

• Parallel angles
• 2D shapes
• Triangles
• Directly proportional graph

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