Geometry Theorems — Isosceles Triangle
Geometry Theorems — Isosceles Triangle. Practice questions to deepen understanding of the isosceles triangle. Online math practice with full solutions and step-by-step explanations.
Isosceles triangle practice — equal base angles, angle bisector = altitude = median, 45-45-90 triangle. Theorems and proofs.
Definition of an isosceles triangle, base angles theorem (equal), the converse theorem.
△ Definition:
An isosceles triangle is a triangle that has:
A triangle that has two equal sides
The equal sides are called: legs
The third side is called: base
The angle between the legs: apex angle
The angles at the base: base angles
📐 Fundamental theorem:
In an isosceles triangle, the base angles are:
In an isosceles triangle,
the base angles are equal!
If AB = AC (legs)
then ∠B = ∠C ✓
↔️ Converse theorem:
If a triangle has two equal angles, then:
Equal sides ⇒ equal base angles
The converse:
Equal angles ⇒ the sides opposite them are equal
If ∠B = ∠C
then AB = AC ✓
The triangle is isosceles!
In a triangle:
two equal sides ⇔ two equal angles
(both directions hold!)
📏 Special line:
In an isosceles triangle, the apex angle bisector is also:
In an isosceles triangle,
the apex angle bisector is also:
1. An altitude to the base (perpendicular) ✓
2. A median to the base (bisects it) ✓
Three roles in one!
🔢 Computation:
In an isosceles triangle, the apex angle is 40°. Each base angle is:
Isosceles triangle
Apex angle = 40°
Solution:
Let each base angle = α
The two base angles are equal: ∠B = ∠C = α
Sum of angles in a triangle:
40° + α + α = 180°
40° + 2α = 180°
2α = 140°
α = 70° ✓
🔍 Identification:
In a triangle, two angles are 65° and 65°. The triangle is:
Two angles: 65° and 65°
Conclusion:
Two equal angles!
By the converse theorem:
Equal angles ⇒ the sides opposite them are equal
The triangle is isosceles! ✓
(The third angle: 180° - 65° - 65° = 50°)
📏 Altitude:
The altitude to the base in an isosceles triangle:
The altitude from the apex angle to the base is also a median
that is: it divides the base into two equal parts ✓
If D is the foot of the altitude on BC:
BD = DC
🔢 Computation:
In an isosceles triangle, a base angle is 55°. The apex angle is:
Base angle = 55°
Solution:
The two base angles are equal: 55° + 55° = 110°
Sum of angles in a triangle:
apex angle + 110° = 180°
apex angle = 180° - 110° = 70° ✓
📐 Angle bisector:
The bisector from a base angle in an isosceles triangle:
Only the bisector from the apex angle is both altitude and median
The bisector from a base angle is not necessarily an altitude or a median!
This is a special property only for the apex angle bisector ✓
△ Special case:
An equilateral triangle is also:
Equilateral triangle = all 3 sides equal
If all 3 are equal, then in particular two are equal!
Therefore: an equilateral triangle is also isosceles ✓
(a special case of isosceles)
equilateral ⊂ isosceles ⊂ general triangle
🔍 Identification:
A triangle with sides 5, 5, 8 is:
Check:
1. Triangle inequality:
5 + 5 = 10 > 8 ✓
2. Two sides are equal (5 = 5) ✓
Conclusion: isosceles!
The legs: 5 and 5
The base: 8
✂️ Division:
When the altitude from the apex angle is drawn in an isosceles triangle, what is created?
The altitude from the apex angle:
1. Creates a right angle (90°) ✓
2. Bisects the base ✓
3. Bisects the apex angle ✓
Two congruent right triangles are created!
(SAS congruence)
🔢 Ratio:
In an isosceles triangle, the ratio of the apex angle to a base angle is 2:5. The apex angle is:
ratio of apex angle : base angle = 2:5
Solution:
Let:
apex angle = 2x
each base angle = 5x
Sum:
2x + 5x + 5x = 180°
12x = 180°
x = 15°
apex angle = 2×15° = 30° ✓
each base angle = 5×15° = 75°
↗️ Exterior angle:
In an isosceles triangle, the exterior angle adjacent to a base angle:
Exterior angle = sum of the two non-adjacent interior angles
In an isosceles triangle:
Exterior angle adjacent to a base angle =
apex angle + other base angle ✓
∠exterior = ∠A + ∠B
📏 Median:
A median to a leg in an isosceles triangle:
The special property (altitude=median=bisector)
holds only for the line from the apex angle to the base!
A median to a leg is not necessarily an altitude or a bisector ✗
It has no special property
📐 Triangle type:
If in an isosceles triangle the apex angle is 110°, the triangle is:
Analysis:
110° > 90° ⇒ obtuse angle!
One obtuse angle exists ⇒ obtuse triangle ✓
Base angles:
each = (180° - 110°)/2 = 35°
(both are acute)
⊿ Special case:
In a right isosceles triangle, each acute angle is:
Right triangle (one angle = 90°)
and isosceles (two equal sides)
Solution:
The angles are: 90°, α, α (the two base angles are equal)
90° + α + α = 180°
90° + 2α = 180°
2α = 90°
α = 45° ✓
A 45-45-90 triangle!
🔢 Computation:
In an isosceles triangle ABC (AB=AC), ∠A=50°. What is ∠B?
AB = AC (legs)
∠A = 50° (apex angle)
Solution:
∠B = ∠C (base angles are equal)
Let ∠B = ∠C = x
50° + x + x = 180°
50° + 2x = 180°
2x = 130°
x = 65° ✓
🔍 Identification:
In a triangle, the three angles are 80°, 50°, 50°. The triangle is:
Analysis:
1. Sum: 80° + 50° + 50° = 180° ✓
2. Two equal angles (50° = 50°) ✓
Conclusion:
By the converse theorem:
two equal angles ⇒ the triangle is isosceles!
apex angle = 80°
base angles = 50° each
📚 Summary:
Which of the following statements about an isosceles triangle is false?
"A median to a leg is always an altitude"
This is false! ✗
Only the line from the apex angle to the base has the special properties (bisector=altitude=median)
A median to a leg is not necessarily an altitude! ⚠️
✓ Base angles are equal
✓ Apex angle bisector = altitude = median to base
✓ Two equal angles ⇒ isosceles
✓ Two equal sides ⇒ base angles are equal