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Converse Of BPT And Applications

Apply the converse of the Basic Proportionality Theorem to prove lines are parallel.

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Concept

Converse of BPT (Theorem 6.2)

Statement of the converse theorem.

Theorem Statement

Theorem 6.2 states: "If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side."

This is known as the Converse of the Basic Proportionality Theorem (BPT). While BPT gives us proportional sides from parallel lines, the converse lets us prove lines are parallel by verifying proportional sides.

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IMAGE

Converse of BPT and Applications Visual

Visuals are needed to show intersecting lines and the parallel outcome.

A clean scientific geometric diagram of a triangle labeled ABC. A solid line DE intersects side AB at point D and side A…

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Worked Example

Example 2: Trapezium Application

Example 2 from page 11.

Problem Statement

Problem. $ABCD$ is a trapezium with $AB \parallel DC$ . Points $E$ and $F$ lie on non-parallel sides $AD$ and $BC$ respectively, such that $EF$ is parallel to $AB$ .

Goal: Show that $\frac{AE}{ED} = \frac{BF}{FC}$ .

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Worked Example

Example 3: Converse in Action

Example 3 from page 11.

Problem Statement

Problem. In $\Delta PQR$ , points $S$ and $T$ lie on sides $PQ$ and $PR$ respectively. We are given the ratio $\frac{PS}{SQ} = \frac{PT}{TR}$ and that $\angle PST = \angle PRQ$ .

Goal: Prove that $\Delta PQR$ is an isosceles triangle.

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Practice

Checking Parallelism

Exercise 6.2 Q2(i).

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In $\Delta PQR$ , points $E$ and $F$ lie on $PQ$ and $PR$ . If $PE = 3.9$ , $EQ = 3$ , $PF = 3.6$ , and $FR = 2.4$ , is $EF \parallel QR$ ?

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Practice Problem

Prove the relationship

Exercise 6.2 Q4 proof problem.

In a given

\Delta ABC

DE \parallel AC

and

DF \parallel AE

. Using these two facts, prove that

\frac{BF}{FE} = \frac{BE}{EC}

. Provide the core theorem used and structure your step-by-step reasoning below.

Which primary theorem forms the basis of your proof?*

Identify the main geometric principle you will apply to relate the parallel lines to side ratios.

Step-by-step Proof*

Write out your logical sequence. Be sure to specify which triangle you are applying the theorem to in each step.

Real-world Application (Optional)

Think of a practical scenario where guaranteeing parallel structures using proportional measurements is useful.

← previousThe Basic Proportionality Theorem next →Similarity Criteria: AAA And AA

Worked Example

Example 2: Trapezium Application

Example 2 from page 11.

Problem Statement

Problem. $ABCD$ is a trapezium with $AB \parallel DC$ . Points $E$ and $F$ lie on non-parallel sides $AD$ and $BC$ respectively, such that $EF$ is parallel to $AB$ .

Goal: Show that $\frac{AE}{ED} = \frac{BF}{FC}$ .

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Worked Example

Example 3: Converse in Action

Example 3 from page 11.

Problem Statement

Problem. In $\Delta PQR$ , points $S$ and $T$ lie on sides $PQ$ and $PR$ respectively. We are given the ratio $\frac{PS}{SQ} = \frac{PT}{TR}$ and that $\angle PST = \angle PRQ$ .

Goal: Prove that $\Delta PQR$ is an isosceles triangle.

1 / 4

Practice

Checking Parallelism

Exercise 6.2 Q2(i).

1 / 5

In $\Delta PQR$ , points $E$ and $F$ lie on $PQ$ and $PR$ . If $PE = 3.9$ , $EQ = 3$ , $PF = 3.6$ , and $FR = 2.4$ , is $EF \parallel QR$ ?

Practice Problem

Prove the relationship

Exercise 6.2 Q4 proof problem.

In a given

\Delta ABC

DE \parallel AC

and

DF \parallel AE

. Using these two facts, prove that

\frac{BF}{FE} = \frac{BE}{EC}

. Provide the core theorem used and structure your step-by-step reasoning below.

Which primary theorem forms the basis of your proof?*

Identify the main geometric principle you will apply to relate the parallel lines to side ratios.

Step-by-step Proof*

Write out your logical sequence. Be sure to specify which triangle you are applying the theorem to in each step.

Real-world Application (Optional)

Think of a practical scenario where guaranteeing parallel structures using proportional measurements is useful.