Question

section 5.7 homework page 2

1. copy and complete the flowchart to show that \\( \overline{ln} \parallel \overline{rd} \\).

given: midsegment \\( ln \\) in \\( \triangle foa
midsegment \\( rd \\) in \\( \triangle ioa
show: \\( \overline{ln} \parallel \overline{rd}
flowchart proof

1. \\( \triangle foa \\) with midsegment \\( ln \\) given
2. \\( \overline{ln} \parallel \overline{oa} \\) ?
3. ? two lines parallel to the same line are parallel.
4. \\( \triangle foa \\) with midsegment \\( rd \\) given
5. ? triangle midsegment conjecture
6. calculate the measure of each lettered angle. explain how you determined the measures \\( h \\) and \\( k \\).

\\( \ell_1 \parallel \ell_2

1. cart is an isosceles trapezoid. what are the coordinates of point \\( t \\)?
2. hse is a kite. what are the coordinates of point \\( r \\)?

Explanation:

Step1: Analyze the Triangle Midsegment Conjecture

The Triangle Midsegment Conjecture states that the midsegment of a triangle is parallel to the third side. For $\triangle FOA$ with midsegment $LN$, by this conjecture, $LN \parallel OA$. Similarly, for $\triangle IOA$ with midsegment $RD$, applying the same conjecture, we get $RD \parallel OA$.

Step2: Use the Transitive Property of Parallel Lines

If two lines ($LN$ and $RD$) are both parallel to the same line ($OA$), then by the transitive property of parallel lines, $LN \parallel RD$.

For the flowchart:

• Step 3: Justification is "Triangle Midsegment Conjecture" (since midsegment $LN$ in $\triangle FOA$ implies $LN \parallel OA$ by the conjecture).
• Step 4: For $\triangle IOA$ with midsegment $RD$, by the Triangle Midsegment Conjecture, $RD \parallel OA$.
• Step 5: Since $LN \parallel OA$ and $RD \parallel OA$, then $LN \parallel RD$ (two lines parallel to the same line are parallel).

Answer:

Flowchart Completion:

• Step 3: Justification - Triangle Midsegment Conjecture (so $LN \parallel OA$)
• Step 4: $RD \parallel OA$ (by Triangle Midsegment Conjecture, as $RD$ is midsegment of $\triangle IOA$)
• Step 5: $LN \parallel RD$ (two lines parallel to the same line $OA$ are parallel)