Lesson 9
Drawing Triangles (Part 1)
Let’s see how many different triangles we can draw with certain measurements.
9.1: Which One Doesn’t Belong: Triangles
Which one doesn’t belong?
![Four triangles, A, B, C, D.](https://cms-im.s3.amazonaws.com/Dxmgvx2xj8RhYRKnTC9iYyXP?response-content-disposition=inline%3B%20filename%3D%227-7.7.9.Image.Revision.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.9.Image.Revision.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141951Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=25e435e3fa92aa78a69514d7608c13f3b763567d0522f0a3e0b37a0c5bbca392)
9.2: Does Your Triangle Match Theirs?
Three students have each drawn a triangle. For each description:
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Drag the vertices to create a triangle with the given measurements.
- Make note of the different side lengths and angle measures in your triangle.
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Decide whether the triangle you made must be an identical copy of the triangle that the student drew. Explain your reasoning.
Jada’s triangle has one angle measuring 75°.
Andre’s triangle has one angle measuring 75° and one angle measuring 45°.
9.3: How Many Can You Draw?
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Draw as many different triangles as you can with each of these sets of measurements:
- Two angles measure \(60^\circ\), and one side measures 4 cm.
- Two angles measure \(90^\circ\), and one side measures 4 cm.
- One angle measures \(60^\circ\), one angle measures \(90^\circ\), and one side measures 4 cm.
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Which sets of measurements determine one unique triangle? Explain or show your reasoning.
![Nine toothpicks are arranged to make 3 equilateral triangles, their bases form a horizontal line.](https://cms-im.s3.amazonaws.com/fk6nQRbpUmLV4NDM8VHWUwWb?response-content-disposition=inline%3B%20filename%3D%227-7.7.ext.3triangles.toothpick.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.ext.3triangles.toothpick.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141951Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=90149f1da0524b3ca6041a97567e9ea8a2a9c59ca945b35f6417a36decc251cf)
In the diagram, 9 toothpicks are used to make three equilateral triangles. Figure out a way to move only 3 of the toothpicks so that the diagram has exactly 5 equilateral triangles.
Summary
Sometimes, we are given two different angle measures and a side length, and it is impossible to draw a triangle. For example, there is no triangle with side length 2 and angle measures \(120^\circ\) and \(100^\circ\):
![Figure of a horizontal line segment with dashed line segments forming angles at each end.](https://cms-im.s3.amazonaws.com/oDPFnMTMcNxw9j8Yk4zEFzJn?response-content-disposition=inline%3B%20filename%3D%227-7.6.B4.Image.09.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B4.Image.09.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141951Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a48817585db36455db3a98f94f1c5747d7429336ce2c737360e7b646b14cca84)
Sometimes, we are given two different angle measures and a side length between them, and we can draw a unique triangle. For example, if we draw a triangle with a side length of 4 between angles \(90^\circ\) and \(60^\circ\), there is only one way they can meet up and complete to a triangle:
![A segment 4 units long is drawn. A dotted line is drawn perpendicular to one end of the segment, a dotted line is drawn at a 60 degree angle to the other end of the segment. The dotted lines meet.](https://cms-im.s3.amazonaws.com/hmCFEsF4a7AVpMxLxF5UAvpL?response-content-disposition=inline%3B%20filename%3D%227-7.6.B4.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B4.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T141951Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a4eb0ee58d9fc9b6f891d8245f8a2562cb402df6ae74e3404d72334e97f180a2)
Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and same angle measures.