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$\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$? I could not get how to use the condition $AD=BC$ , How do I use the condition to find $\angle{DBC}$? EDIT 1: With MvG's observation, we can prove the following fact.
First, we will show if we set a point $E$ on the segment $AC$ such that $OE=OB=OC=BC$, then $D=E$. Becuase $\triangle{ABC}$ is a isosceles triangle, the point $O$ is on the bisecting line of $\angle{BAC}$. $\angle{OAE}=20^\circ/2=10^\circ$. And because $OE=OC$, $\angle{OCE}=\angle{OEC}=20^\circ$, $\angle{EOA}=20^\circ-10^\circ=10^\circ=\angle{EAO}$. Therefore $\triangle{AOE}$ is an isosceles triangle such that $EA=EO$. so $AD=BC=AE$, $D=E$. Now we can see the point $O$ is a circumcenter of the $\triangle{DBC}$ because $OB=OC=OD.$ By using this fact, we can find $\angle{DBC}=70^\circ$,
A triangle with two sides of equal length is an isosceles triangle. Examples of Isosceles Triangle:Not an Isosceles Triangle:Examples of Isosceles Triangles in Real Life:Many things in the world have the shape of an isosceles triangle. Some popular examples of these triangles in real life are: Parts of an Isosceles TriangleParts of an isosceles triangle 1. Legs: The two equal sides of an isosceles triangle are known as ‘legs’. In the triangle ABC (given above), AB and AC are the two legs of the isosceles triangle. 2. Base: The ‘base’ of an isosceles triangle is the third and unequal side. In the triangle ABC, BC is the base of the isosceles triangle. 3. Vertex angle: The ‘vertex angle’ is the angle formed by two equal sides of an isosceles triangle. ∠BAC is a vertex angle of the isosceles triangle. 4. Base angles: The ‘base angles’ are the angles that involve the base of an isosceles triangle. ∠ABC and ∠ACB are the two base angles of the isosceles triangle. Properties of an Isosceles TriangleHere is a list of some properties of isosceles triangles:
In the isosceles triangle given above, the two angles ∠B and ∠C, opposite to the equal sides AB and AC are equal to each other.
Types of Isosceles TrianglesGenerally, isosceles triangles are classified into three different types:
Area and Perimeter of Isosceles Triangle
Area (A) = ½ × base (b) × height (h)
Perimeter (P) = 2a + base (b) Here, ‘a’ refers to the length of the equal sides of the isosceles triangle and ‘b’ refers to the length of the third unequal side. Solved ExamplesExample 1What is the height of an isosceles triangle with an area of 12 sq. cm and a base of 6 cm? Solution: Area of isosceles triangle = ½ x base x height i.e. 12 = ½ x 6 x height i.e. 12 = 3 x height i.e. height = 4 cm Example 2What is the perimeter of an isosceles triangle, if equal sides are ‘a’ cm each and the unequal side is ‘b’ cm? Solution: Perimeter of an isosceles triangle = sum of its sides Perimeter of an isosceles triangle = (a + a + b) cm, i.e., (2a + b) cm Example 3Find the perimeter of an isosceles triangle if the base is 16 cm and the equal sides are 24 cm each. Solution: Formula of the perimeter of an isosceles triangle, P = 2a + b Here, a (sides) = 24 cm and b (base) = 16 cm Therefore, perimeter of an isosceles triangle, P = 2(24) + 16 = 64 cm. Hence, the perimeter is 64 cm. Triangle GamesWith SplashLearn, there are several games about triangles for children to try. Let us look at a few of them:
Other Games
Students may also find it a bit overwhelming to remember the properties of isosceles triangles. But that’s precisely where you will require a great deal of patience while teaching your kid. Allow your child to shine bright with SplashLearn. Practice Problems on Isosceles TrianglesAttend this Quiz & Test your knowledge. Correct answer is: 4 cm Correct answer is: AC ≠ BC Correct answer is: 45 cm2 Frequently Asked Questions
How do we know if a triangle is isosceles?
A triangle is said to be an isosceles triangle if any of its two sides are equal. Let’s take a triangle that has AB, BC, and CA as its three sides. If any of these are true—AB = BC, BC = CA or CA = AB—then the triangle is isosceles.
Can a right triangle also be an isosceles triangle?
Yes, a right triangle or right-angle triangle can be an isosceles triangle. An isosceles right triangle will have 1 right angle and 2 other angles as equal angles.
Can you find all the angles of an isosceles triangle if you know one of the equal angles?
Yes, if we know the two equal angles, then we can easily subtract the sum of it from 180°, since the sum of all angles of a triangle is equal to 180°.
What are some of the properties of an isosceles triangle?
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