### Clarifications

*Clarification 1:*Problems include using the Triangle Sum Theorem and representing angle measures as algebraic expressions.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Geometric Reasoning

**Standard:**Develop an understanding of the Pythagorean Theorem and angle relationships involving triangles.

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Angle (∠)
- Supplementary Angles

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grades 4 and 5, students used angle measures as an attribute of two-dimensional figures and identified and classified angles as acute, right, obtuse, straight and reflex. In grade 8, students solve problems involving the interior and exterior angles of a triangle. In Geometry, the work in this benchmark will be extended to proving relationships and theorems about triangles.- Students should explore the concept before being provided the theorem. Once conceptual understanding is developed, students can use numerical equations to solve problems involving finding missing interior or exterior angles. From there, students should develop algebraic equations to solve for both missing angle measurements as well as variables.
- Instruction includes students exploring and using deductive reasoning to determine relationships that exist between angle sums and exterior angle sums of triangles. Students should construct various triangles and find the measures of both the interior and exterior angles. Applying knowledge of these relationships, students can use deductive reasoning to find the measure of any missing angles.
- Using an investigation, such as the one below, for the Triangle Sum Theorem will help students conceptually understand the total degree measures of a triangle related to a straight line.
- Give each student a triangle (a variety of triangle sizes will allow for discussion).
- Have students use a lined sheet of paper or draw a straight line using a ruler to represent the straight angle of 180°.
- Next, have the students tear off each angle of the triangle.
- Then, putting the angle side towards the line, the students will be able to model the 180° of the triangle measures to a straight line. This is illustrated in the top part of the figure below.
- A similar investigation can be used to connect the measure of an exterior angle of a triangle and the two opposite interior angles in the triangle.

- Students can also use patty paper to trace the concepts to make connections.

### Common Misconceptions or Errors

- Students may incorrectly think that the exterior angle is the whole reflex angle of the interior angle.
- Students may not recognize that there are two exterior angles for each interior angle and that the exterior angles are congruent.

### Strategies to Support Tiered Instruction

- Teacher provides a visual with one angle of the triangle on a straight line of 180°, similar to the illustration below. Teacher provides instruction on how to measure each of the angles with a protractor and color code congruent angles.
- Teacher provides tracing paper or patty paper to trace the related interior and exterior angles and have discussion about the angle relationships.
- Instruction includes providing a visual with one angle of the triangle on a straight line of 180° and measuring each angle in the figure. Teacher co-constructs a graphic organizer with students to document the angle measurements, color-code congruent angles in the figure and in the graphic organizer, and identifies the relationships between interior and exterior angles of a triangle and the Triangle Sum Theorem.

### Instructional Tasks

*Instructional Task 1*

**(MTR.1.1, MTR.2.1)**Δ$A$$B$$C$ and Δ$B$$C$$D$ share a common side of $B$$C$. Angle $B$$A$$C$ is 30° and angle $A$$B$$C$ is 60°.

- Part A. Create a diagram to represent this description.
- Part B. What will be the measure of angle $B$$C$$D$? Provide an explanation to support how to find the measurement of angle $B$$C$$D$.
- Part C. Are there questions that still need to be answered to approach Part B?

### Instructional Items

*Instructional Item 1*

In triangle $L$$M$$N$, the measure of angle $L$ is 50° and the measure of angle $M$ 70°. What is the measure of the exterior angle to angle $N$?

*Instructional Item 2*

One measure of an angle in a triangle is 96°. The other two angle measures are represented by 2$x$ and $x$ + 12. Determine the other two degree measures for the missing angles.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessment

## Lesson Plans

## Problem-Solving Tasks

## Tutorial

## MFAS Formative Assessments

Students are asked to apply the Exterior Angle of a Triangle Theorem and provide an informal justification.

## Student Resources

## Problem-Solving Tasks

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task

## Tutorial

In this video, we find missing angle measures from a variety of examples.

Type: Tutorial

## Parent Resources

## Problem-Solving Tasks

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task