Answer:
see attached
Explanation:
You want to draw a triangle with angle measures 30° and 75° at vertices Y and Z, and the length of YZ as 8.5 cm.
Third angle
The third angle of the triangle will be 180° -30° -75° = 75°. This means the triangle is isosceles, and the two congruent sides are both 8.5 cm.
Protractor
Once you measure the length YZ as 8.5 cm, you can draw angles at 30° and 75° from points Y and Z, respectively. A protractor is a suitable tool for the purpose. The point of intersection of these rays is X, giving you the desired triangle XYZ.
Construction
A right triangle with hypotenuse 8.5 cm will have the leg opposite the 30° angle as 4.25 cm, half the hypotenuse. This offers opportunities to construct the triangle several ways.
Here's one. Draw an arc (or circle) of radius 8.5 cm centered at point Y. Mark a point Z on the arc. Using the same radius, draw an arc that intersects the first one at point A. Angle ZYA is 60°, as is angle YZA. Now, you can bisect angle ZYA, or segment ZA, or you can simply measure off 4.25 cm on segment ZA. The point halfway between Z and A can be marked point B. Where ray YB intersects the original arc (or circle) centered at Y, is point X of your triangle XYZ.
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Additional comment
We used a geometry program to draw the triangle in the attachment. It conveniently provided angles with the necessary measures. In terms of the construction described above, the vertex of the right angle is point B.
Another way to start is by identifying right angle lines BZ and BY on grid paper. Once you locate point Z 4.25 cm from B, you can draw an arc centered at Z with radius 8.5 cm that intersects line BY at Y. Then an arc with the same radius centered at Y will identify the location of point X.