MA.912.T.1.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.GR.1.2
• MA.912.GR.1.6 
• MA.912.GR.6.3 
• MA.912.GR.7.2
• MA.912.GR.7.3

Terms from the K-12 Glossary

• Angle 
• Triangle 

Vertical Alignment

Previous Benchmarks

• MA.8.GR.1.1
• MA.8.GR.1.2
• MA.8.GR.2.4

Next Benchmarks

• MA.912.T.1.5
• MA.912.T.1.6
• MA.912.T.1.7
• MA.912.T.1.8
• MA.912.NSO.3.5
• MA.912.NSO.3.9

Purpose and Instructional Strategies

• Within the Geometry course, the expectation is to use angle measures given in degrees and not in radians. Additionally, it is not the expectation for students to master the trigonometric ratios of secant, cosecant and cotangent within this course.
• It is customary to use Greek letters to represent angle measures (e.g., θ, α, β, γ).
• The Law of Sines states that given a triangle with side lengths $a$, $b$ and $c$, and angle measures α, β and γ (opposite to the sides respectively), then $\frac{\text{<math><mi>a</mi></math>}}{\text{sin α}}$ = $\frac{\text{<math><mi>b</mi></math>}}{\text{sin β}}$ = $\frac{\text{<math><mi>c</mi></math>}}{\text{sin γ}}$ or $\frac{\text{sin α}}{\text{<math><mi>a</mi></math>}}$ = $\frac{\text{sin β}}{\text{<math><mi>b</mi></math>}}$ = $\frac{\text{sin γ}}{\text{<math><mi>c</mi></math>}}$.
• Instruction includes making the connection between the Law of Sines and the congruence criteria of Angle-Side-Angle and Angle-Angle-Side. 
  • For example, when given two angle measures and the included side length, students can use the Law of Sines to determine the missing side lengths and angle measures and justify that there is one unique triangle as indicated by the Angle-Side-Angle congruence criterion. 
• Instruction includes making the connection between the Law of Sines and the Side-Side-Angle criterion, which is referred to as the ambiguous case. 
  • For example, when given two side lengths and an angle measure of an opposite angle students can use the Law of Sines to determine the value of the sines of the missing angle measures. If the value of either of the sines is greater than one, then no triangle exists with those angle measures. If the value of both of the sines is less than one, then there are two possible triangles, one acute and one obtuse. If the value of either of the sines is equal to one, then there is exactly one triangle, a right triangle, with those measures. 
• The Law of Cosines states that given a triangle with side lengths $a$, $b$ and $c$, and angle measures α, β and γ (opposite to the sides respectively), then $a$² = $b$² + $c$² − 2$b$$c$ cos α, $b$² = $a$² + $c$² − 2$a$$c$ cos β and $c$² = $a$² + $b$² − 2$a$$b$ cos γ. 
• Instruction includes making the connection between the Law of Cosines and the congruence criteria of Side-Side-Side and Side-Angle-Side. 
  • For example, when given three side lengths, students can use the three versions of the Law of Cosines to determine the missing angle measures and justify that there is one unique triangle as indicated by the Side-Side-Side congruence criterion. However, if the side lengths do not satisfy the Triangle Inequality Theorem, then no triangle exists. 
• Instruction includes making the connection between the Law of Cosines and the Side-Side-Angle criterion, which is referred to as the ambiguous case. 
  • For example, when given two side lengths and an angle measure of an opposite angle students can use one of the versions of the Law of Cosines to obtain a quadratic equation for the missing side length. If there is no real solutions, then no triangle exists. If there are two real solutions, then there are two possible triangles, one acute and one obtuse. If there is exactly one real solution, then there is exactly one triangle, a right triangle. 
• Instruction includes the understanding that the Pythagorean Theorem is a special case of the Law of Cosines when the included angle measure is 90°, whose cosine is zero. (MTR.5.1) 
• Instruction includes the understanding that sines and cosines of obtuse angles are necessary when solving problems involving obtuse triangles. Enrichment of this benchmark includes the connection to the unit circle.

Common Misconceptions or Errors

• Students may have trouble keeping track of which sides or angles belong in the Laws of Sines and Cosines.

Instructional Tasks

• Triangle ABC is an acute triangle. The measure of AB is 17.8 inches, the measure of BC is 16.5 inches and the measure of angle C is 75°.
  
  
  
  • Part A. Determine the measure of angle A. Compare your method used with a partner.
  • Part B. Determine the measure of angle B.

• Part A. Given side lengths 10, 11 and 22, use the Law of Cosines to determine if a triangle exists with those side lengths.
• Part B. How could you determine, without using the Law of Cosines, whether such a triangle exists?
• Part C. If the triangle from Part A does not exist, change one of its side lengths so that a triangle does exist.

Instructional Items

• Find the remaining measures of the angles and sides of triangle ABC.

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