Question

name - asa angles - sides - (triangle with vertices l, k, m; another triangle with vertex h and side marked with three equal signs)

Explanation:

Step1: Identify Triangle Parts

The triangle has vertices \( J \), \( K \), \( M \). Angles at \( K \) and \( M \) (let's assume they are right angles? Wait, no, the left triangle has a right angle. Wait, maybe we need to label angles and sides. Let's see: the table has columns Name - ASA, Angles, Sides. For triangle \( \triangle JKM \), let's define angles: \( \angle K \), \( \angle M \), \( \angle J \). Sides: \( JK \), \( KM \), \( JM \). Wait, maybe the left triangle is \( \triangle H \) with a right angle and equal sides (marked with three lines, so congruent sides). But the main triangle is \( \triangle JKM \). Let's list:

- **Name (ASA):** For ASA (Angle - Side - Angle), we need two angles and the included side. Suppose \( \angle K \) and \( \angle M \) are angles, and \( KM \) is the included side. But maybe the angles at \( K \) and \( M \) are equal? Wait, the left triangle has a right angle and congruent sides (isosceles right triangle). Maybe the right triangle \( \triangle JKM \): let's assume \( \angle K \) and \( \angle M \) are angles, \( KM \) is side \( n \) (wait, left triangle has side \( n \) with three marks, so congruent to another side). Maybe the triangle \( \triangle JKM \) has angles \( \angle K \), \( \angle M \), and \( \angle J \), sides \( JK \) (length \( n \)?), \( KM \) (length \( n \)?), \( JM \). Wait, maybe the left triangle is \( \triangle H \) with right angle, two congruent sides (so isosceles right triangle, angles 45 - 45 - 90). Then \( \triangle JKM \): if \( \angle K \) and \( \angle M \) are 45 degrees, and \( KM \) is the included side, then ASA would be two angles (45, 45) and included side \( KM \). But maybe the problem is to fill the table. Let's structure:

- **Angles:** Let's say \( \angle K \), \( \angle M \), \( \angle J \). If it's an isosceles triangle (from left triangle's congruent sides), maybe \( \angle K = \angle M \), and \( \angle J \) is the vertex angle.
- **Sides:** \( JK \), \( KM \), \( JM \). If \( JK = KM \) (like left triangle), then two sides are equal.

Wait, maybe the table is for triangle \( \triangle JKM \):

| Name - ASA | Angles | Sides |
| --- | --- | --- |
| \( \triangle JKM \) | \( \angle K \), \( \angle M \), \( \angle J \) | \( JK \), \( KM \), \( JM \) |

But maybe the left triangle is \( \triangle H \) with right angle, so \( \angle H = 90^\circ \), and two congruent sides (so \( \angle H \) is right angle, sides \( H \) (vertical) and horizontal are congruent, so angles 45 - 45 - 90. Then \( \triangle JKM \): if \( \angle K \) and \( \angle M \) are 45°, and \( KM \) is the included side, then ASA is \( \angle K \), \( KM \), \( \angle M \).

Wait, maybe the problem is to identify the angles and sides for the triangle using ASA. Let's clarify:

- **Angles:** Let's assume \( \angle K \) and \( \angle M \) are the two angles, and \( KM \) is the included side (ASA). So angles: \( \angle K \), \( \angle M \), and the included side \( KM \).
- **Sides:** \( JK \), \( KM \), \( JM \). If \( JK \) and \( KM \) are congruent (like left triangle), then \( JK = KM \).

But maybe the left triangle is \( \triangle H \) with right angle, so \( \angle H = 90^\circ \), sides \( H \) (vertical) and horizontal are congruent (so length \( n \)), hypotenuse \( \sqrt{2}n \). Then \( \triangle JKM \): if \( \angle K = \angle M = 45^\circ \), \( \angle J = 90^\circ \), sides \( JK = KM = n \), \( JM = \sqrt{2}n \).

So filling the table:

- **Name (ASA):** \( \triangle JKM \) (ASA with \( \angle K \), \( KM \), \( \angle M \))
- **Angles:** \( \angle K = 45^\circ \), \( \angle M = 45^\circ \), \( \angle J = 90^\circ \)
- **Sides:** \( JK = n \), \( KM = n \), \( JM = n\sqrt{2} \)

But maybe the problem is to list angles and sides for the triangle. Let's confirm:

- Angles: In a triangle, sum to \( 180^\circ \). If two angles are \( 45^\circ \), the third is \( 90^\circ \).
- Sides: In an isosceles right triangle, legs are equal, hypotenuse is leg \( \times \sqrt{2} \).

So the table would have:

| Name - ASA | Angles | Sides |
| --- | --- | --- |
| \( \triangle JKM \) | \( \angle K = 45^\circ \), \( \angle M = 45^\circ \), \( \angle J = 90^\circ \) | \( JK = n \), \( KM = n \), \( JM = n\sqrt{2} \) |

Answer:

For the triangle (assuming isosceles right triangle as per left triangle's congruent sides and right angle):

• Angles: \( \angle K = 45^\circ \), \( \angle M = 45^\circ \), \( \angle J = 90^\circ \)
• Sides: \( JK = n \), \( KM = n \), \( JM = n\sqrt{2} \)

(Note: The "Name - ASA" would be \( \triangle JKM \) with ASA criteria met by \( \angle K \), \( KM \), \( \angle M \).)