Question

triangle knm is isosceles, where angle n is the vertex. what is the measure of angle k? 11° 25° 50° 65°

Explanation:

Step1: Set angles equal (isosceles triangle)

In an isosceles triangle with vertex \( N \), the perpendicular from \( N \) to \( KM \) (at \( L \)) bisects \( KM \) and makes \( \angle KNL = \angle MNL \). So, \( 5x + 10 = 6x - 1 \).
Solve for \( x \):
\( 6x - 5x = 10 + 1 \)
\( x = 11 \)

Step2: Find \( \angle KNL \)

Substitute \( x = 11 \) into \( 5x + 10 \):
\( 5(11) + 10 = 55 + 10 = 65^\circ \)? Wait, no, wait. Wait, \( \angle KNL \) is \( 5x + 10 \), but \( \angle K \) is complementary to \( \angle KNL \) because \( NL \perp KM \), so \( \angle KLN = 90^\circ \). Wait, no, wait. Wait, in triangle \( KNL \), it's a right triangle? Wait, \( NL \) is perpendicular to \( KM \), so \( \angle KLN = 90^\circ \). So in triangle \( KNL \), angles sum to \( 180^\circ \). Wait, but first, we found \( x = 11 \), so \( \angle KNL = 5(11) + 10 = 65^\circ \)? Wait, no, that can't be. Wait, no, wait: \( 5x + 10 = 6x - 1 \) gives \( x = 11 \), so \( \angle KNL = 5(11) + 10 = 65^\circ \)? Wait, no, that would make \( \angle K \) equal to \( 90^\circ - 65^\circ = 25^\circ \)? Wait, no, wait. Wait, let's re-examine.

Wait, the triangle is isosceles with \( KN = MN \) (since \( N \) is the vertex and \( NL \) is the perpendicular bisector). So \( \angle K = \angle M \). Also, \( NL \perp KM \), so \( \angle KLN = \angle MLN = 90^\circ \). The angles \( \angle KNL \) and \( \angle MNL \) are equal (because \( NL \) is the angle bisector in the isosceles triangle). So we set \( 5x + 10 = 6x - 1 \), solve \( x = 11 \). Then \( \angle KNL = 5(11) + 10 = 65^\circ \). Then in right triangle \( KNL \), \( \angle K + \angle KNL + 90^\circ = 180^\circ \), so \( \angle K = 180^\circ - 90^\circ - 65^\circ = 25^\circ \)? Wait, no, that's not right. Wait, no, \( \angle KNL \) is \( 5x + 10 \), but when \( x = 11 \), \( 5(11) + 10 = 65 \), but then \( \angle K \) would be \( 90 - 65 = 25 \). Wait, but let's check the answer options. The options are \( 11^\circ \), \( 25^\circ \), \( 50^\circ \), \( 65^\circ \). So \( 25^\circ \) is an option. Wait, but let's do it again.

Step 1: Solve \( 5x + 10 = 6x - 1 \)
\( 6x - 5x = 10 + 1 \)
\( x = 11 \)

Step 2: Find \( \angle KNL \): \( 5(11) + 10 = 65^\circ \)

Step 3: In right triangle \( KNL \), \( \angle K + \angle KNL + 90^\circ = 180^\circ \)
So \( \angle K = 180^\circ - 90^\circ - 65^\circ = 25^\circ \)

Yes, that makes sense. So \( \angle K = 25^\circ \).

Answer:

\( 25^\circ \) (corresponding to the option \( 25^\circ \))