Chapter 5: Triangle Constructions (part 2: sides & angles)

#4) How to Copy an Angle

To copy an angle, follow the steps below: 1) Create a circle centered at A. 2) Label the points where the arc crosses each side as "X" and "Y." ("Intersection" tool to find intersection points, right click to rename points). Let "X" be on the horizontal side. 3) Use the "compass" tool to copy the circle, this time centered at A'. Find and label X'. 4) To find Y' and create the second side of the angle, use the "compass" tool to create a circle centered at X' with radius = XY. 5) Identify and label Y', then place a ray from A' through Y'. You should now have an exact copy of the original angle!

Move around the yellow points to change the size of angle A. Notice that angle A' will always be congruent to angle A. How does this construction process work? In other words, explain/prove why angle P and angle P' are always congruent. Hint: add segments between X&Y and X'&Y'. What can you say about triangle AXY and triangle A'X'Y'? Why?

#5a) Testing for Angle-Side-Angle ("ASA") Congruence

Goal: Let's try to copy a triangle by copying two angles and the included side. Step 1: Copy the included side. - Use the "COMPASS" tool to make a circle with radius = AB, centered at A'. - Use the "POINT" tool to pick any point on the circle. Rename (right click) this point as B'. - Use the "SEGMENT" tool to connect A' and B'. - Now you have a side whose length will always equal AB. Step 2: Copy the angle on one side. - Use the steps from #4 to copy angle A to vertex A'. Step 3: Copy the angle on the other side. - Use the steps from #4 to copy angle B to vertex B'. Step 4: Find and label the third point. - Find and label the intersection of the rays as point C'. Step 4: "lighten" the arc marks. - Click on any arc or segment. A color & line menu should appear in the top right. You can use it to change the color, thickness, and/or dotted-ness of your work. Now, the three sides should stand out visually, but the circles are still visible as a way to show your work.

Move points A, B, and C around to produce different types of triangles. As you move these points, triangle A'B'C' will also change. Using inductive reasoning (testing many different possible triangles), do you think "ASA" always determines a triangle? In other words, if two triangles have two sets of congruent corresponding angles, and the included sides are congruent, must the triangles be congruent to one another?

#5b) Testing the Limits of ASA

Try to construct a triangle using "ASA" from the components below, by copying angle B onto the given angle and horizontal side.

Why can't you construct a triangle in this case? Would changing the length AB make any difference?

What must be true about the given angle measures in order for ASA to determine a triangle?

#6) Testing for Side-Angle-Side ("SAS") Congruence

Goal: Let's try to copy a triangle by copying two side and the included angle. Step 1: Copy one side. - Use the "COMPASS" tool to make a circle with radius = AB, centered at A'. - Use the "POINT" tool to pick any point on the circle. Rename (right click) this point as B'. - Use the "SEGMENT" tool to connect A' and B'. - Now you have a side whose length will always equal AB. Step 2: Copy the included angle. - Use the steps from #4 to copy angle A to vertex A'. Step 3: Copy the other side. - Use the "COMPASS" tool to make a circle with radius = AC, centered at A'. - Label and rename point C'. Step 4: Fill in the third side. - Use the "SEGMENT" tool to add the remaining side. Step 4: "lighten" the arc marks. - Click on any arc or segment. A color & line menu should appear in the top right. You can use it to change the color, thickness, and/or dotted-ness of your work. Now, the triangle should stand out visually, but the circles are still visible as a way to show your work.

Move points A, B, and C around to produce different types of triangles. As you move these points, triangle A'B'C' should also change. Consider potential "counterexample" cases, like making angle A is obtuse. Using inductive reasoning (testing many different possible triangles), do you think "SAS" always determines a triangle?

Chapter 5: Triangle Constructions (part 2: sides & angles)

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