# Triangles

Triangles are closed shapes having three $$3$$ angles, three sides, as well as three vertices. Triangles with $$3$$ vertices say $$P, Q,$$ and $$R$$ are characterized as $$△PQR$$. It’s additionally called a $$3$$-sided polygon or a trigon.

Crucial properties of triangles are shown here:

• Triangles have$$3$$ sides, angles, and vertices.
• The angle total property of a triangle says the amount of the $$3$$ inner triangles is constantly $$180°$$. Like with any particular triangle $$PQR$$, the angle $$P +$$ angle $$Q +$$ angle $$R = 180°$$.
• Triangle’s inequality property says the amount of the two sides’ length for triangles is bigger than the $$3$$rd side.
• Like in the Pythagorean theorem, with a right triangle, the square of the hypotenuse equates to the quantity of the squares of the additional $$2$$ sides such as $$(Hypotenuse² = Base² + Altitude²)$$.
• The side which is opposite the larger angle is the one that is the longest.
• The outer angle’s property of a triangle says the outer triangle angle always is equivalent to the total of the inside opposite angles.

Triangles can be classified based on angles and sides:

## Right Triangles

A right angle’s definition says if one of the triangle’s angles is a right angle – $$90º$$, it’s known as a right-angled triangle or merely, a right triangle.

Several critical properties characterize and assist in identifying right triangles.

• The biggest angle is constantly $$90º$$.
• The biggest side is known as a hypotenuse, and constantly it’s the side opposite of a right angle.
• The Pythagoras rule governs the dimensions of the sides.
• It can’t contain an obtuse angle.

## Acute triangles:

Acute triangles are those classified based on the angles’ measurements. Should every inner angle in the triangle be lower than $$90°$$, it’s an acute triangle.

Acute Angle Triangles’ properties are:

• Based on the angle’s sum property, all $$3$$ inner angles of the acute triangle combine to form $$180°$$.
• Triangles can’t be both right-angled triangles and acute-angled triangles all at once.
• Triangles can’t acute-angled triangles as well as obtuse-angled triangles all at once.
• The angle’s property of an acute triangle declares the inner angles of acute triangles are constantly fewer than $$90°$$ or are in-between ($$0°$$ to $$90°$$).
• The side which is opposite to the tiniest angle is the tiniest triangle side.

## Obtuse Triangles:

Within geometry, obtuse scalene triangles are defined as triangles with one angle measuring over $$90$$ degrees, yet lower than $$180$$ degrees, plus the additional $$2$$ angles are a smaller amount than $$90$$ degrees. All $$3$$ sides, as well as the angles, vary in length.

Properties of obtuse scalene triangles are:

• They have $$2$$ acute angles as well as $$1$$ obtuse angle.
• All its sides and angles are distinct in measurement.
• The total of all $$3$$ inner angles equal $$180°$$.

### Triangles – Example 1:

Find the measure of the unknown angle in the triangle

Solution:

The sum of the inner angles of a triangle is $$180°$$ . So, $$90°$$$$+$$$$45^°$$$$=$$$$135^°$$$$→$$$$180^°$$$$-$$ $$135^°$$ $$=$$ $$45^°$$ . The unkown angle is $$45^°$$

### Triangles – Example 1:

Find the measure of the unknown angle in the triangle

Solution:

The sum of the inner angles of a triangle is $$180°$$ . So, $$120°$$$$+$$$$35^°$$$$=$$$$155^°$$$$→$$$$180^°$$$$-$$ $$155^°$$ $$=$$ $$25^°$$ . The unkown angle is $$25^°$$

## Exercises for Triangles

Find the measure of the unknown angle in each triangle.

1)

2)

• $$\color{blue}{70^°}$$
• $$\color{blue}{30^°}$$

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