On a blank sheet of paper, sketch Angle GHI. Then sketch JK parallel to GH such that point J is on GI and point K is on HI. Prove, using a two-column proof or a paragraph proof, that GJ/JI=HK/KI.
To prove that GJ/JI=HK/KI, we can use similar triangles.
First, let's sketch Angle GHI on a blank sheet of paper. It's essential to label the points correctly while sketching.
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Using ASCII art to sketch Angle GHI
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K/_________________\J
G H
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In the sketch above, we have angle GHI, where G, H, and I are the vertices. We will draw a line parallel to GH passing through point J and K such that J lies on GI, and K lies on HI.
Now, let's label the points:
G(x₁, y₁),
H(x₂, y₂),
I(x₃, y₃),
J(x₄, y₄),
K(x₅, y₅).
We need to prove that GJ/JI=HK/KI, so we will show that the triangles GJI and HKI are similar.
Proof:
1. We know that JK is parallel to GH, by construction.
2. By the parallel lines intersected by a transversal theorem, the corresponding angles are equal:
- Angle JGH = Angle JKJ (alternate interior angles)
- Angle GHJ = Angle JKH (corresponding angles)
3. Since GJI is a triangle, the sum of the angles is 180 degrees. So, we have:
- Angle JGI = 180 - Angle GJI - Angle GJ (angle sum property)
- Using the values from step 2, Angle JGI = 180 - Angle GHI - Angle JKH (substituting the angles)
4. Similarly, we can find Angle HKI:
- Angle HKI = 180 - Angle KHI - Angle HK (angle sum property)
- Using the values from step 2, Angle HKI = 180 - Angle GHI - Angle JKH (substituting the angles)
5. From step 3 and step 4, we can see that Angle JGI = Angle HKI.
6. By the Angle-Angle (AA) similarity criterion, we have the similarity of triangles GJI and HKI.
7. In similar triangles, the ratio of corresponding sides is equal, so:
- GJ/JI = HK/KI
Therefore, we have proved that GJ/JI=HK/KI using the two-column proof above.
The proof in part 1b demonstrates that JK divides two sides of GHI proportionally because it shows that GJ/JI is equal to HK/KI.
In the two-column proof or paragraph proof, we established that triangles GJI and HKI are similar. When two triangles are similar, the corresponding sides are in proportion.
So, since GJI and HKI are similar triangles, the ratio of corresponding sides is equal. This means that we can say GJ/JI is equal to HK/KI.
Therefore, we can conclude that JK divides the sides of triangle GHI proportionally, with the ratio of GJ/JI being equal to the ratio of HK/KI.
Explain how your proof in part 1b demonstrates that JK divides two sides of GHI proportionally.
To prove GJ/JI=HK/KI, we need to show that the corresponding sides are proportional. Let's begin with the given information:
1. Sketch Angle GHI on a blank sheet of paper.
2. Sketch a line JK parallel to GH, where J is on GI and K is on HI.
Now, let's start the proof:
Statement | Reason
--- | ---
1. GJ‖HK | Given (Parallel Lines)
2. ∠GJH ≅ ∠HKI | Corresponding Angles of Parallel Lines
3. ∠IJG ≅ ∠IKH | Corresponding Angles of Parallel Lines
4. △GJI ≅ △HKI | Angle-Angle Similarity (AA)
5. GJ/JI = HK/KI | Corresponding Parts of Similar Triangles are Proportional
Explanation of each step:
1. It is given that line GJ is parallel to line HK since both are parallel to GH.
2. This statement follows from the nature of parallel lines. When two parallel lines are intersected by another line, the corresponding angles formed are congruent.
3. Similarly, the corresponding angles formed by the intersected lines and parallel lines are congruent.
4. Since ∠GJH ≅ ∠HKI (as shown in statement 2) and ∠IJG ≅ ∠IKH (as shown in statement 3), the two triangles are similar by the Angle-Angle (AA) similarity postulate.
5. By the corresponding parts of similar triangles, we know that the ratios of the lengths of the corresponding sides are equal. Hence, GJ/JI = HK/KI.
Therefore, we have proved that GJ/JI=HK/KI using a two-column proof.