Geometry Proofs

Question

1. The ratio of the volume of a pyramid to the volume of a prism with the same base and height is _____ 

Because the formula for the volume of a prism is V=Bh , where B is the area of the base and h is the height, the formula for the volume of a pyramid is _____

 

              (possible answers)     3 to 4    1 to 3    1 to 2    1 to 1    V=3/4Bh    V=1/3Bh    V=1/2Bh   V=Bh

 

2. Given: Quadrilateral DEFG is inscribed in circle P.

 

Prove: m∠G+m∠E=180∘ 

 

https://i.gyazo.com/3c8d1b810d766d6c6fce10dbe4f80aa4.png

 

It is given that quadrilateral DEFG is inscribed in circle P. Because a circle measures 360°, (arc)mDEF+(arc)mFGD=360∘ . By the _____ , 1/2(arc)mDEF+1/2(arc)mFGD=180∘ . By the inscribed angles theorem, _____ = 1/2(arc)mFGF and _____ = 1/2m(arc)DEF. This means m∠G+m∠E=180∘ by the _____

 

       (possible answers)     m∠D    m∠E    m∠F    m∠G    addition property of equality    multiplication property of equality    transitive property of equality    substitution property of equality 

 

m∠
m∠
m∠
m∠

Answer 1

1) The formula for the volume of pyramid with same base and height is Bh/3. So ratio of volume of the pyramid and the volume of prism is

(Bh/3)/Bh = 1/3. So possible ans is 1 to 3

2) To prove m<G+m<E = 180 degree, use inscribed quadrilateral theorem which states that a quadrilateral can be inscribed in circle if and only if the opposite angles of a quadrilateral are supplementary.

Inscribed angle E has half the measurements of intercepted arc DGF

Intercepted arc DGF is 180 degree because full circle is 360 degree

So angle E is 90 degree

Similarly we can prove that inscribed angle G has half the measurements of intercepted arc DEF which is 180 degree so

angle G is 90 degree

Therefore m<E+m<E = 90+90 = 180 degree