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GPS Alignment: Instructor Page
Week 11: Geometry - Triangles


Half As Much May Be Right (Printable PDF)

What is significant about an inscribed (blue) angle in a semi-circle? Why might this fact be true?

 

GPS

As seen in Problem Exploration

M6G1. Students will further develop their understanding of plane figures.

Students can use GSP to construct the triangle in the semicircle and then measure the degree of the angle. The students will develop a better understanding of inscribed angles. Also, the following properties are needed: supplementary angles on a straight angle, triangle angle sum is 180 degrees, isosceles triangles have two congruent sides, and base angles of an isosceles triangle are congruent..
 

M7G1. Students will construct plane figures that meet given conditions.

a. Perform basic constructions using both compass and straight edge, and appropriate technology. Constructions should include copying a segment; copying and angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
 

Students will construct the semi-circle and inscribed angle either with a compass and straight edge or GSP. Students can then measure the angle. If using GSP, the students can dynamically move the point along the semi-circle. If using a compass and straight-edge, the student can position the angle in various locations on the semi-circle and measure the angle.
 

M7A1 Students will represent and evaluate quantities using algebraic expressions.

b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.

c. Add and subtract linear expressions.
 

In solving this problem algebraically, the students can divide the inscribed triangle into two separate triangles. Let the angles for the two triangles be as follows: <1,<2,<3 for one triangle, and <4,<5,<6 for the other with (<1 + <4) being equal to the inscribed angle (blue). Using various properties of triangles (see M8G1) and supplementary angles, the following algebraic equations can be set up for the angle sum of the triangles:
<1 + <2 + <3 = 180
<4 + <5 + <6 = 180

Since we are dealing with isosceles triangles, the base angles are congruent (so we can substitute one for another in the above equations)
2(<1) + <3 = 180
2(<4) + <6 = 180

Adding the equations together and using the fact that supplementary angles add to 180, we eventually get <1+<4 = 90
 

M8G1. Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.

 

The students will construct two isosceles triangles in this investigation, which can lead to a discussion of the idea of congruence. Once the students work through the problem, they will find that two of the sides are perpendicular.

M8A1 Students will use algebra to represent, analyze, and solve problems.

b. Simplify and evaluate algebraic expressions.

c. Solve algebraic equations in one variable, including equations involving absolute values.

Same as M7A1.