Unit 7 Angles Triangles And Prisms — Unit Plan

1. $28^\circ$
2. $53^\circ$

1.  $ABC$ with one of $BAC$, $FAD$, or $ADG$
2. One of these pairs:
   1. $CAD$ with one of $ABC$, $DAC$, or $DAG$
   2. $BAF$ with one of $ABC$, $DAC$, or $DAG$
   3. Any 2 of $BCA$, $ACG$, and $CGA$
3. One of these pairs:
   1. $DAF$ and $BAC$
   2. $BAF$ and $CAD$

1. Angle $BOC=80^\circ$. Sample reasoning: Given angle $DOF=120^\circ$, angle $AOC=120^\circ$ because they are congruent vertical angles. Consequently, angles $AOB+BOC=120^\circ$ because they are adjacent.
2. Angle $COD=60^\circ$. Sample reasoning: Angle $COD$ and angle $DOF$  are supplementary angles, so the sum of their measurements has to be $180^\circ$.

1. Samples responses: $2w + 76 = 180$ or $4w + 152 = 360$.
2. Sample responses:
   • Subtract 76 from 180 and then divide by 2 (or multiply by $\frac12$).
   • Subtract 152 from 360 and then divide by 4 (or multiply by $\frac14$).

1. Yes, Jada’s shape has 4 sides, all measuring 5 cm.
2. A rhombus could be made with all four sides the same length, but without right angles.

1. Elena should put a point where the two circles intersect and draw line segments connecting that point to points $A$ and $B$ to finish her triangle.
2. Elena’s drawing means that there is no way to draw a triangle with these side lengths. The circles do not intersect, because the side lengths of 3 inches and 4 inches are too short to make a triangle with the third side of 8 inches.

1. These are both the same triangle. In both cases, the $60^\circ$ angle is between the 3-cm and 5-cm sides. If you trace one triangle, flip it and turn it, it can line up exactly with the other triangle.
2. To draw a different triangle, they should try putting the $60^\circ$ angle next to the side of unknown length, instead of between the two known sides.

1. Yes. Sample reasoning: Diego’s triangle has two $45^\circ$ angles and a side length of 8 cm.
2. Yes, there is another possible triangle. Sample reasoning: Diego could keep one $45^\circ$ angle next to the 8-cm side, but move the other one across from the 8-cm side.

1. Noah should use a compass to draw a circle with radius 6 cm and center at one end of the 4-cm side. He should then draw segments connecting both ends of the 4-cm side to the point where the circle and ray cross, and that will complete the triangle.
2. Yes. Noah could try beginning with the same setup he has already drawn again, but this time center the circle on the other end of the 4-cm side. He could also start with the 6-cm side drawn instead of the 4-cm side and follow the same process.

1. Cross-sections:
   1. A pentagon with all sides the same length, but smaller than the base of the pyramid
   2. A triangle
2. Sample responses:
   1. You could slice the pyramid diagonally.
   2. You could slice the pyramid vertically but not through the top vertex.

1. 287 in³, because the base has an area of 41 in^2, and $41\boldcdot 7=287$.
2. 3 in, because $41 \boldcdot 3 = 123$.

The volume is 232 cm³. The area of the base is 29 cm² and can be found in multiple ways, but one way is to consider a 5 by 7 rectangle with a right triangle cut off, then $5 \boldcdot 7 - \frac{1}{2} \boldcdot 4 \boldcdot 3 = 29$. Since the height is 8 cm, the volume is calculated by $29 \boldcdot 8 = 232$.

The surface area is 270 cm². Possible strategy: The area of the base is 27 cm². The perimeter of the base is 24 cm, so the combined area of the sides is 216 cm², because $24 \boldcdot 9=216$. Therefore the total surface area is 270 cm², because $27 \boldcdot 2 + 216=270$.

Sample response: Volume refers to how much of something fits inside an object. Surface area refers to how much of something is needed to cover the outside of an object.

1. 75 square feet. $(2.5 \boldcdot4)+2(5 \boldcdot4)+2(2.5 \boldcdot5)=75$
2. 2 bottles of paint. $\frac {75}{40}=1.875$