special properties of isosceles triangles

SPECIAL PROPERTIES OF ISOSCELES TRIANGLES HOW TO

Theorem #2 (converse) – If two angles of a triangle are congruent, the sides opposite them are congruent. The converse of this theorem looks at the reverse. Notice that the base of the triangle is created by both angles that are congruent. The arcs that are in the angles are indicating that the angles have the same measure, or are congruent. The single lines on the legs of the isosceles triangles are tick marks indicating that the sides have the same length, or are congruent. To find the opposite angle you want to look at the angle that the side is not a part of. This means that if we know that two sides are congruent in a triangle, we know that two angles are congruent as well. Isosceles Triangle Theorems Theorem #1 – If two sides of a triangle are congruent, the angles opposite them are congruent.

SPECIAL PROPERTIES OF ISOSCELES TRIANGLES HOW TO

Now let’s see how to find the missing sides of an isosceles triangle and calculate their lengths. Knowing that an isosceles triangle has two sides that are equal leads us to the first theorem that is associated with isosceles triangles. Oftentimes a problem will use this vocabulary to give information. The two sides that are the same length are referred to as the legs and the third side is called the base. Often abstract or complicated shapes are broken into smaller shapes such as a triangle. Many triangles found in the real world can be considered isosceles, including a section of a slice of pizza. Isosceles Triangle Properties, Characteristics & Uses These words make a difference when considering if a triangle is also an equilateral triangle as well as an isosceles triangle. QU is an angle bisector of Δ QRS because it bisects ∠ RQS.Definition: An isosceles triangle is defined as a triangle having two congruent sides or two sides that are the same length.Īn isosceles triangle can also be an equilateral triangle, but it doesn’t have to be.ĭefinitions for these triangles typically include the word “only” or “exactly”. SP is a median to base QR because P is the midpoint of QR. RT is an altitude to base QS because RT ⊥ QS. In Figure, the altitude drawn from the vertex angle of an isosceles triangle can be proven to be a median as well as an angle bisector.įigure 9 The altitude drawn from the vertex angle of an isosceles triangle.Įxample 1: Based on the markings in Figure 10, name an altitude of Δ QRS, name a median of Δ QRS, and name an angle bisector of Δ QRS.įigure 10 Finding an altitude, a median, and an angle bisector. In certain triangles, though, they can be the same segments. In general, altitudes, medians, and angle bisectors are different segments. In every triangle, the three angle bisectors meet in one point inside the triangle (Figure 8).įigure 8 The three angle bisectors meet in a single point inside the triangle. In Figure, is an angle bisector in Δ ABC.

Every triangle has three angle bisectors. In every triangle, the three medians meet in one point inside the triangle (Figure 6).įigure 6 The three medians meet in a single point inside the triangle.Īn angle bisector in a triangle is a segment drawn from a vertex that bisects (cuts in half) that vertex angle. Which may or may not be inside the triangle.Ī median in a triangle is the line segment drawn from a vertex to the midpoint of its opposite side. It is interesting to note that in any triangle, the three lines containing the altitudes meet in one point (Figure 4).įigure 4 The three lines containing the altitudes intersect in a single point, įigure 3 An altitude for an obtuse triangle. In Figure 3, AM is the altitude to base BC. įigure 2 In a right triangle, each leg can serve as an altitude.

In Figure 2, AC is an altitude to base BC, and BC is an altitude to base AC. Every altitude is the perpendicular segment from a vertex to its opposite side (or the extension of the opposite side) (Figure 1).įigure 1 Three bases and three altitudes for the same triangle.Īltitudes can sometimes coincide with a side of the triangle or can sometimes meet an extended base outside the triangle. Now isn't that kind of special?Įvery triangle has three bases (any of its sides) and three altitudes (heights). Just as there are special names for special types of triangles, so there are special names for special line segments within triangles.

Summary of Coordinate Geometry Formulas.

Slopes: Parallel and Perpendicular Lines.

Similar Triangles: Perimeters and Areas.

Proportional Parts of Similar Triangles.

Formulas: Perimeter, Circumference, Area.

Proving that Figures Are Parallelograms.

Triangle Inequalities: Sides and Angles.

Special Features of Isosceles Triangles.Classifying Triangles by Sides or Angles.Lines: Intersecting, Perpendicular, Parallel.

Special properties of isosceles triangles