How do you prove the converse of the triangle proportionality theorem?

The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. Proving — Converse of the Triangle Proportionality Theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Procedure

  1. Cut an acute-angled triangle say ABC from a coloured paper.
  2. Paste the ΔABC on ruled sheet such that the base of the triangle coincides with ruled line.
  3. Mark two points P and Q on AB and AC such that PQ || BC.
  4. Using a ruler measure the length of AP, PB, AQ and QC.

Secondly, what is SAS Similarity Theorem? SAS Similarity Theorem: If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, then the triangles are similar.

Herein, how do you find the similarities of a triangle?

Triangles are similar if:

  1. AAA (angle angle angle) All three pairs of corresponding angles are the same.
  2. SSS in same proportion (side side side) All three pairs of corresponding sides are in the same proportion.
  3. SAS (side angle side) Two pairs of sides in the same proportion and the included angle equal.

What is the side splitter Theorem?

The “Side SplitterTheorem says that if a line intersects two sides of a triangle and is parallel to the third side of the triangle, it divides those two sides proportionally. If 2 || lines are cut by a transversal, the corresponding angles are congruent. 3.

What is the triangle Midsegment Theorem?

A line segment that connects two midpoints of the sides of a triangle is called a midsegment. The Midsegment Theorem states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side.

Are parallel lines similar?

Parallel Lines and Similar and Congruent Triangles Theorem 6.1: If two parallel lines are transected by a third, the alternate interior angles are the same size. Theorem 6.2: If a line intersects two other lines then the following conditions are equivalent. a) The alternate interior angles are the same size.

What are congruent triangles?

Congruent Triangles. When two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there.

What is the area theorem?

In the mathematical theory of conformal mappings, the area theorem gives an inequality satisfied by the power series coefficients of certain conformal mappings. The theorem is called by that name, not because of its implications, but rather because the proof uses the notion of area.

What is the converse of Pythagorean Theorem?

The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

What is converse Thales Theorem?

Converse of Basic Proportionality Theorem: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

Which is basic proportionality theorem?

Basic Proportionality Theorem (can be abbreviated as BPT) states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion.

How does the Pythagorean theorem work?

The Pythagorean theorem deals with the lengths of the sides of a right triangle. The theorem states that: The sum of the squares of the lengths of the legs of a right triangle (‘a’ and ‘b’ in the triangle shown below) is equal to the square of the length of the hypotenuse (‘c’).

How do you prove the MidPoint theorem?

MidPoint Theorem Proof Let E and D be the midpoints of the sides AC and AB. Then the line DE is said to be parallel to the sides BC, whereas the side DE is half of the side BC; i.e. DE = (1/2 * BC). Construction- Extend the line segment DE and produce it to F such that, EF=DE.