Hinge Theorem

Front

What is the Hinge Theorem?

Back

The Hinge Theorem states that if two triangles have two sides of one triangle equal to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the side opposite the larger angle in the first triangle is longer than the side opposite the smaller angle in the second triangle.

Front

How does the Hinge Theorem apply to comparing sides of triangles?

Back

The Hinge Theorem allows us to determine which side of two triangles is longer based on the angles between the equal sides. If one triangle has a larger included angle, its opposite side will be longer.

Front

If triangle ABC has sides AB = 5, AC = 7, and angle A = 60°, and triangle DEF has sides DE = 5, DF = 7, and angle D = 30°, which side is longer?

Back

Side AC is longer than side DF because angle A is larger than angle D.

Front

What is the relationship between angles and sides in triangles according to the Hinge Theorem?

Back

In triangles, the larger the angle, the longer the side opposite to it. This relationship is crucial for applying the Hinge Theorem.

Front

If two triangles have two equal sides and one triangle has a larger included angle, what can be concluded about the third side?

Back

The third side of the triangle with the larger included angle will be longer than the third side of the triangle with the smaller included angle.

Front

What is an example of using the Hinge Theorem in real life?

Back

An example is determining which of two ladders will reach higher when both are placed against a wall at different angles.

Front

In triangle XYZ, if XY = 10, XZ = 12, and angle X = 70°, and in triangle PQR, if PQ = 10, PR = 12, and angle P = 50°, which side is longer?

Back

Side XZ is longer than side PR because angle X is larger than angle P.

Front

What is the converse of the Hinge Theorem?

Back

The converse states that if two triangles have two sides equal and the third sides are unequal, then the triangle with the longer third side has the larger included angle.

Front

How can the Hinge Theorem help in construction?

Back

It helps in ensuring that structures are built with the correct angles and lengths, ensuring stability and safety.

Front

If triangle ABC has sides AB = 8, AC = 6, and angle A = 45°, and triangle DEF has sides DE = 8, DF = 6, and angle D = 30°, which side is longer?

Back

Side AC is longer than side DF because angle A is larger than angle D.