d The smaller triangle is half the size of the big triangle.
A
aSimilar? Yes.
Explanation: SAS Similarity Theorem Two-Column Proof: See solution.
B
b ∠ IFJ ≅ ∠ GFH
∠ HGF ≅ ∠ JIF
∠ FHG ≅ ∠ FJI
C
c Yes, they are. See solution.
D
d x=4
GH=13 units
Practice makes perfect
a If the triangles are similar, they should have at least two pairs of congruent angles. Examining the diagram, we see that △ FGH and △ FIJ share an angle at F. Therefore, these angles are congruent according to the Reflexive Property of Congruence. Let's add this information to the diagram and also label the length of the congruent sides.
Let's separate the triangles. Notice that FI is twice the length of FG and FJ is twice the length of FH.
If △ FIJ~ △ F'GH, the ratio of the two pairs of corresponding sides that form the included angle at F should be equal. Let's investigate that.
a/2a? =b/2b ⇔ 1/2=1/2
With this information we can claim similarity by the SAS (Side-Angle-Side) Similarity Theorem. Let's show this as a two-column proof.
Statement
Reason
1.
& FG ≅ GI & FH≅ HJ
1.
Given
2.
FG/FI=FH/FJ
2.
Ratio of corresponding sides are equal
3.
△ FGH ≅ △ FIJ
3.
SAS Similarity Theorem
b From Part A, we know that ∠ IFJ ≅ ∠ GFH. We also know that FI and FG are corresponding sides, as are FJ and FH. With this information we can identify the remaining congruent angles.
Let's summarize what we found.
∠ IFJ ≅ ∠ GFH
∠ HGF ≅ ∠ JIF
∠ FHG ≅ ∠ FJI
We could also claim this by using the fact that ∠ FHG ≅ ∠ FJI.
d From previous parts we know that △ FGH ~ △ FIJ. We also know that △ FGH is half the size of △ FIJ. Therefore, GH must be half the length of IJ. We can use this to write an equation.
2GH=IJBy substituting the expressions for GH and IJ, we can solve for x.
